3$ case). Remarkably, it is possible to prove that ${\cal L}< -E <0$ without a detailed study of the function $f_\gamma$, but only using equation (43). Indeed, one first shows that $\cal L$ is self-adjoint and that $-E_\gamma$, its largest eigenvalue, satisfies $-E_\gamma\leq -E_{\gamma_p}$. So, writing $E \equiv E_{\gamma_p}$, it is enough to show that $-E <0$. Next, it is easy to see, using the Feynman-Kac formula \cite{Si} and the the Perron-Frobenius theorem \cite{GJ} that ${\cal L}$ has a unique eigenvector $\Omega$ with eigenvalue $-E$ and $\Omega$ can be chosen to be strictly positive. Write (43) as $$ {\cal L} f = -(p-1) f^p. $$ So, $$ (\Omega, {\cal L} f) = -(p-1) (\Omega,f^p) $$ where $(\cdot,\cdot)$ denotes the scalar product in $L^2 ({\Bbb R},d\mu)$. By the self-adjointness of ${\cal L}$ and the definition of $\Omega$, i.e. ${\cal L}\Omega = -E\Omega$, we have $$ -E = -(p-1) \frac{(\Omega,f^p)}{(\Omega,f)} < 0 $$ since $\Omega$ and $f$ are strictly positive. This proves our claim. Notice that functions in $L^2 ({\bf R}, d\mu)$ have essentially a Gaussian decay at infinity, which is much faster than what is allowed in our Banach space $B$, see (47). An extra work is needed, using the Mehler's formula for the kernel of $e^{\tau{\cal L}_0}$ (see \cite{BK5} for details). Finally, let us express this result in the RG language. The RG map has a fixed point ${ R}_{L}(f_\gamma,F^*)=(f_\gamma,F^*)$ for $F^*(u)=-u|u|^{p-1}$. There are different fixed points for different values of $\gamma$, but since the decay of the functions in $B$ is faster than the one of any of these fixed points, each of them is unique in the corresponding set $\{f_\gamma + h\}$ for $h$ as in Theorem 2. In other words, unlike the situation for $p>3$, we do not have to deal with a line of fixed points and here there is no constant like $A$ in Theorem 1. This is reflected in the fact, (53), that the linear semigroup $e^{\tau{\cal L}}$ contracts. \vs{2mm} \no {\bf Remark.} The reader who is familiar with the theory of critical phenomena will recognize an analogy between the behaviour of an Ising model at its critical point, when the dimension of the lattice changes and the asymptotic behaviour of the solution of (41) when $p$ varies. This is further discussed in \cite {BKL}. Here, our results for $p<3$ are non-perturbative, i.e. we do not use any ``$\varepsilon$-expansion" (see \cite{Wa} for an analysis of the bifurcation at $p=3$). Also, the reader may notice that our approach to the renormalization group is close to the ``Wilsonian" one in the theory of critical phenomena, while the method developed in \cite{Go} is analogous to the renormalized perturbation of quantum field theory. For a rigorous treatment of Barenblatt's equation, which was one of the first example analyzed in \cite{Go}, see \cite{KP3}. \subsection{Higher-order asymptotics.} It is interesting to reexamine the $p\geq 3$ case, using what we learned from the analysis of $p<3$. This has been done by Wayne \cite{Wa}. Namely, consider equation (15), and make the change of variable (48), but replacing $f_\gamma$ by 0, i.e. \BE u(x,t) = t^{-\frac{1}{p-1}} v(xt^{-1/2},\log t) \EN Then $v(\xi,\tau)$ satisfies \BE v_\tau = {\cal L}_0 v + \lambda v^p \EN with ${\cal L}_0$ as in (51); we have computed the spectrum of ${\cal L}_0$ which is $ \{ \frac{1}{p-1} - \frac{1}{2} - \frac{m}{2} | m=0,1,\cdots\}. $ Now, notice that, for $p>3$, this spectrum is entirely negative. Thus $e^{\tau{\cal L}_0}$ contracts exponentially and it is not hard to show that, for $\lambda$ small, and $v(\xi,0)$ bounded in a suitable norm, the same contraction holds for the solution of (56). Actually, since the largest eigenvalue of ${\cal L}_0$ corresponds to $m=0$, we expect a decay like $e^{\tau(\frac{1}{p-1}-\frac{1}{2})}$, which, using $\tau = \log t$ and combining it with the $t^{-\frac{1}{p-1}}$ prefactor in (55), leads to a $t^{-1/2}$ decay of $u(x,t)$, for all $p\geq 3$. This of course has to be the case, in view of Theorem 1. Actually, much more can be said. Take the $n$ largest eigenvalues $\{\lambda_1,\cdots,\lambda_n\}$ of ${\cal L}_0$ and the corresponding subspaces. Then, using a theorem of Gallay \cite{Gal2}, Wayne constructs an invariant manifold (in the $v$ variable) such that any solution (for $p>3$ and small initial data) will converge to that invariant manifold at a rate at least of order $t^{-(\lambda_{n+1}-\delta)}$ (with, as before, $\delta$ arbitrarily small for sufficiently small initial conditions). Since, for $n=1$, the eigenvector of ${\cal L}_0$ is $e^{-\xi^2/4}$, one easily recovers Theorem 1. But all higher order asymptotics in time of $v(\xi,\tau)$, hence of $u(x,t)$, can also be obtained in terms of Hermite functions (such higher order corrections were also derived in \cite{Li}). For $p<3$, an invariant manifold can still be constructed but it becomes unstable $(\frac{1}{p-1} - \frac{1}{2} >0)$ and thus, it does not, in general, give information on the long-time asymptotics. Of course, we know from Theorem 2 what the situation is: instead of 0, we have to consider the solution $f_\gamma$ and, presumably, a similar picture (with invariant manifolds and higher order asymptotics) holds there, since the spectrum of the operators ${\cal L}$ for $p<3$ is qualitatively similar to the one of ${\cal L}_0$ for $p>3$. \subsection{An application to reaction-diffusion equations.} In \cite{BX}, Berlyand and Xin consider the following model which leads to a nice RG analysis and which exhibits the novel feature of anomalous exponents depending on the initial data: \BE u_t &=& \Delta u + v u^{p-1} \\ v_t &=& \Lambda^{-1} \Delta v - v u^{p-1} \EN where $u=u(x,t), v = v (x,t), x \in {\bf R}^n$ (we take $n=1$ for simplicity) and $\Lambda > 0$ is the Lewis number. Equations (57,58) model a chemical reaction $A \to B$ where $v$ is the mass fraction of reactant $A$ and $u$ the one of reactant $B$. First, observe that for $p>3$, the analysis done for (15) applies and the asymptotics of $u,v$ is Gaussian. So let $p=3$, so that the nonlinear terms are marginal, and let us see heuristically what happens. If $u$ and $v$ are positive (which is physically necessary and is preserved by (57,58) due to the maximum principle) then, again by the maximum principle, $u$ is larger than the solution of the heat equation with the same initial data: \BE u(x,t) \geq \frac{A}{t^{1/2}} e^{-\frac{x^2}{4t}}. \EN where $A$ depends on $u(x,0)$. Then, $v$ is less than the solution of (58) with $u^{p-1} = u^2$ replaced by its lower bound (59). Calling $\bar v$ this upper bound on $v$, $\bar v$ solves a linear equation \BE \bar v_t = \Lambda^{-1} \Delta \bar v - A^2 t^{-1} e^{-\frac{x^2}{2t}} \bar v. \EN It is not hard to see, by direct substitution, that (60) admits a self-similar solution of the form \BE \bar v (x,t) = t^{-\frac{\alpha}{2}} f^*_\alpha (\frac{x}{\sqrt t}). \EN Indeed, writing $f^*_\alpha (\xi) = e^{-\frac{\Lambda \xi^2}{8}} \Psi (\xi)$, $\Psi$ solves \BE - \Lambda^{-1} \Psi'' + (\frac{\Lambda}{16} \xi^2 + \frac{A^2}{4 \pi} e^{- \xi^2/2} + \frac{1}{4}) \Psi = \frac{\alpha}{2} \Psi \EN We see that, requiring $f^*_\alpha$ to be positive means that $\Psi$ is the ground state of the operator in the LHS of (62) i.e. of an harmonic oscillator perturbed by a positive, rapidly decaying, potential. The exponent $\frac{\alpha}{2}$ is the corresponding ground state energy. For the harmonic oscillator, $\frac{\alpha}{2} = \frac{1}{2}$ and, perturbation theory tells us that, for $A$ small, \BE \frac{\alpha}{2} = \frac{1}{2} + \frac{A^2}{4 \pi \sqrt{2\Lambda^{-1}+1}} + h.o.t. \; \mbox{in} \; A \; > \; \frac{1}{2}. \EN Thus, $\alpha$ depends on $A$, i.e. on $u(x,0)$. Since $\alpha > 1$, $\bar v$, and hence $v$, decay strictly faster than the solution of the heat equation. Inserting this in (57) gives an upper solution for $u$, which solves an equation where the nonlinear term is now {\it irrelevant}. These considerations lead us to expect $u$ to have the heat equation decay: \BE u(x,t) \simeq \frac{A}{\sqrt t} e^{-x^2/4t} \EN and $v$ to have the anomalous decay: \BE v(x,t) = \frac{B}{t^{\alpha/2}} f^*_\alpha (\frac{x}{\sqrt t}) \EN where $A,B$ and $\alpha$ depend on the initial data. Upper and lower bounds of the form (64,65) are proven in \cite{BX}. From the RG point of view, one can define a map $$ R_{L,\alpha} (u,v) = (u_L(\cdot,1),v_L(\cdot,1)) $$ where \BE u_L(x,t) &= & Lu(Lx,L^2t), \\ v_L (x,t) &=& L^\alpha v(Lx,L^2t). \EN We have thus a two-parameter family of fixed points $(Af^*_0,B f^*_{\alpha(A)})$, and the scaling exponent varies continuously with $A$. \section{Patterns and fronts.} \setcounter{equation}{0} \vs{5mm} \subsection{The Ginzburg-Landau equation.} In the previous section, we have seen that nonlinear equations with initial data decaying at infinity produce universal, Gaussian or non-Gaussian, diffusive profiles. Here we shall show that other types of asymptotic behaviour can also exhibit such universality. To discuss a concrete example, consider the Ginzburg-Landau equation \be u_t= u_{xx}+u-|u|^2u \en where $u:{\bf R}\times{\bf R}\rightarrow{\bf C}$ is complex. This equation has a two parameter family of stationary solutions \be u_{q\theta}(x)=\sqrt{1-q^2}e^{i(qx+\theta)} \en and a natural question is to inquire about the time development of initial data $u(x)$ which approach two such solutions at $\pm\infty$: \be \lim_{x\rightarrow\pm\infty}|u(x)-u_{q_\pm\theta_\pm}(x)| =0 . \en This problem has been extensively studied for equation (1) with $u$ {\it real} and $u(-\infty)=1$, $u(\infty)=0$ (i.e. $q_-=0$, $q_+=0$) \cite{AW,Br}. Then the solution takes the form of a propagating front. This occurs because $u(x)=1$ is a stable stationary solution, while $u(x)=0$ is an unstable one. For complex $u$, the solutions (2) are stable, under small perturbations, for $q^2< {1\over 3}$ (the Eckhaus stable domain) \cite {CEE}. We shall consider two questions. The first one was suggested in \cite{CE}, namely we take $q_\pm$ in (3) small, belonging to the Eckhaus stable domain, but not necessarily equal. What is then the long-time asymptotics of the solution? The second question concerns the stability of the real front solutions of (1) for real or complex $u$. For reviews on these questions, we refer the reader to \cite{CE1,CE2}. \subsection{Non-Gaussian patterns.} For the first question, we considered in \cite{BK2} a class of initial data satisfying (3) and we showed that, for any interval $I$, \be \sup_{x\in I}|u(x,t)- e^{i\sqrt{t}\phi^*}u_{q^*\theta^*}(x)|\leq {C_I\over\sqrt{t}} \en where the constants $q^*$, $\phi^*$ and $\theta^*$ depend only on the boundary conditions (3). For a more detailed bound, see Section 3 of \cite{BK2}. Graphics of the solution can be found in \cite{CE2}. In order to see where (4) comes from, we write, following \cite{CE}, \be u=(1-s)e^{i\phi} . \en Equation (1) becomes, in these variables, \be s_t&=& s_{xx}-2s+3s^2-s^3+ \phi_x^2-s\phi_x^2\equiv s_{xx}-2s+F(s,\phi_x)\non\\ \phi_t&=& \phi_{xx}-{2\phi_x s_x\over 1-s}\equiv \phi_{xx}+G(s, s_x,\phi_x) \en with the initial data ( taking again $t=1$ as initial time) \be \lim_{x\rightarrow\pm\infty}s(x,1)=s_{\pm}\; ,\;\; \lim_{x\rightarrow\pm\infty}|\phi(x,1)-\phi_\pm x-\theta_\pm| =0 . \en where $2s_\pm=F(s_\pm,\phi_\pm)$. We proved in \cite{BK2} the following asymptotics, as $t\rightarrow\infty$: \be &&\phi(x,t)=\sqrt{t}\phi^*({x\over\sqrt{t}})+ \theta^*({x\over\sqrt{t}})+{\cal O}({1\over\sqrt{t}}) \\&&s(x,t)= s^*({x\over\sqrt{t}})+{1\over\sqrt{t}} r^*({x\over\sqrt{t}})+{\cal O}({1\over t}) \en for a set of initial data satisfying (7), with $\phi_\pm $, and $\theta_\pm$ small enough (see again \cite{BK2} for the precise definition of the norms in which these limits take place). The functions $\phi^*$, $\theta^*$, $s^*$, $r^*$ are {\it universal}, depending on the initial data only through the boundary conditions (7). They are smooth and therefore (expanding $\phi^*$ to first order, and using (2)) the phase of $u$ will have the asymptotics (4), with $$\phi^*=\phi^*(0),\;\;q^*=\phi^{*'}(0),\;\;\theta^*=\theta^*(0).$$ The behaviour of $s$ will be discussed below. The peculiar scaling behaviour exhibited by (8) can already be understood through the linear heat equation. Previously, we considered the Gaussian fixed point of the RG in a space of integrable initial data. Since the heat equation has stationary solutions of the form $a+bx$, we may also consider the solution of $\phi_t = \phi_{xx}$ with initial data $\phi (x,1) = f(x)$ satisfying (7). The solution can be computed: \be t^{-\ha}\phi(\sqrt{t}x,t+1)={1\over \sqrt{4\pi t}}\int e^{-{1\over 4} (y-x)^2} f(\sqrt{t}y)dy,\non\\ \rightarrow_{t\rightarrow\infty} \phi_{-}x+(\phi_{+}-\phi_{-})(x+2{d \over dx})e(x) \equiv \phi^*_0(x) \en where $e(x) = \int^x_{-\infty} e^{- \frac{y^2}{4}} dy$ is the error function. This gives the first term in (8), with $\phi^\ast$ replaced by $\phi^\ast_0$. If we want to obtain the next correction, we write $\phi^\ast = \phi^\ast_0 + \theta$, where $\theta (x,t)$ solves $\theta_t = \theta_{xx}$ with $\theta(x,1) \to \theta_{\pm}$ as $x \to \pm \infty$. One computes that \be \theta(x,t) \to \theta^\ast_0 \left(\frac{x}{\sqrt t}\right) = \theta_- + (\theta_+ - \theta_-) e \left(\frac{x}{\sqrt t}\right) \en and this yields the second term in (8). It is trivial to check that $\phi^\ast_0$ (resp. $\theta^\ast_0$) is a fixed point of the RG map (2.36) with $\alpha = - 1$ (resp. $\alpha = 0$), and $F=0$. We shall explain below that $\phi^\ast, \theta^\ast$ are small perturbations of $\phi^\ast_0, \theta^\ast_0$. However, as we saw, the scaling (8,4) can be understood qualitatively on the basis of the heat equation, and the boundary conditions (7). We still need to understand (9). The latter comes from the ``slaving" of $s$ to $\phi$, due to the linear $-2s$ term in (6). This will imply that $s$ will have the right form so that (4) holds. Concretely, the slaving means that, to leading order, we may solve the algebraic equation $-2s+F(s,\phi_x)=0$, which is equivalent to $1-s = \sqrt{1-( \phi_x)^2}$, and substitute the result in the equation (6) for $\phi$. Since the derivative of $s$ is proportional to the second derivative of $\phi$, we obtain an equation for $\phi$ only, of the form: \be \phi_t=(1+a(\phi_x,\phi_{xx}))\phi_{xx}\; ,\;\;\phi(x,1)=f(x) \en with the boundary conditions (7) for the initial data $f$. The function $a$ is analytic around the origin. The RG map acts again on pairs of initial data and equations and can be defined (with $\alpha=-1$) as \be R_L(\phi,a) = (\phi_L, a_L) \en where \BE \phi_L(x,t)=L^{-1} \phi(Lx,L^2t) \nonumber \EN and \be a_L(u,v)=a(u,L^{-1}v). \en The semigroup property follows by observing that $\phi_L$ satisfies \be \phi_{Lt}=(1+a_L(\phi_{Lx},\phi_{Lxx}))\phi_{Lxx}. \en We may now iterate $R$ as before, i.e. solve a finite time problem, to study the asymptotics of (12). One wants to show that there exist functions $\phi^{\ast}, a^{\ast}$ such that \be R_L^n(\phi, a) \rightarrow (\phi^\ast, a^{\ast}) \en where \be R_L(\phi^\ast, a^{\ast}) =(\phi^\ast, a^{\ast}) \en is the fixed point of the RG, corresponding to the scale-invariant equation $ \phi_t=(1+a^\ast) \phi_{xx}$. Then, replacing ${x\over\sqrt{t}}$ by $\xi$, the asymptotics of the original problem is given by \be \phi(\xi t^{1/2},t)\sim t^{{1\over 2}}\phi^\ast (\xi). \en Because of the factor $L^{-1}$ in the second argument of $a$ in (14), the fixed point equation is \be \phi_t=(1+a^*(\phi_x))\phi_{xx} \en where $a^*(\cdot) =a(\cdot,0)$. The fixed point is the scale invariant solution $ \phi(x,t)=\sqrt{t}\phi^*({x\over\sqrt{t}}). $ We get for $\phi^\ast$ the equation \be (1+a^*){d^2 \over d\xi^2}\phi^* +\ha \xi{d \over d\xi} \phi^*-\ha\phi^*=0 \en with $a^*=a^*({d\over d\xi} \phi^*)$ and, for small $\phi_\pm$, we look for a solution \be \phi^*=\phi^*_0+\rho \en where $\rho (\pm\infty)=0$ and $\phi^*_0$ is the ``Gaussian" solution (10), which solves (20) with $a^*=0$. This is easy to solve by a fixed point argument(see Proposition 1 in \cite{BK2} or the Proposition in Section 4 of \cite{BKL}). This gives us the first term in (8). Turning to $\theta^*$, we write \be \phi(x,t)=\phi^*(x,t)+\theta(x,t) \en where, with an abuse of notation, $\phi^* (x,t)= \sqrt{t} \phi^* (\frac{x}{\sqrt{t}})$ and $\phi^*$ is given by (21), while $\phi$ solves (12). Then, \be \theta_t=\theta_{xx}+(a\phi_{xx}-a^*\phi^*_{xx}) \en with $\theta(\pm\infty)=\theta_\pm$. Now we set \be \theta_L(x',t')=\theta(Lx',L^2t'). \en Putting $x=Lx', t=L^2t'$, we have $\frac{d^l}{dx^l} \phi^* (Lx',L^2t')=L^{1-l} \frac{d^l}{dx'^l} \phi^*(x',t')$ and therefore $\theta_L$, satisfies the equation (replacing $(x',t')$ by $(x,t)$): \be \theta_{Lt}&=&\theta_{Lxx}+L(a(\phi^*_x+L^{-1} \theta_{Lx}, L^{-1}\phi^*_{xx}+L^{-2}\theta_{Lxx}) (\phi^*_{xx}+L^{-1}\theta_{Lxx}) \non\\ &&-a^*(\phi^*_x)\phi^*_{xx}). \en Thus, reasoning as above, we expect \be \theta_{L^n}\rightarrow\theta^* \en where $\theta^*(x,t)=\theta^*({x\over \sqrt{t}})$ satisfies the $L\rightarrow\infty$ form of (25): \be \theta^*_t=\theta^*_{xx}+a^*\theta^*_{xx}+ (a_u(\phi^*_x,0)\theta^*_x+a_v(\phi^*_x,0)\phi^*_{xx})\phi^\ast_{xx}. \en This is a linear equation, easy to solve, whose solution is, for $\theta_\pm$ small, a small perturbation of the ``Gaussian" solution (11) (which solves (27) with $a=0$). Finally, one sets (with the same abuse of notation) \be \phi(x,t)=\phi^*(x,t)+\theta^*(x,t)+\psi(x,t)=\sqrt{t}\phi^* ({x\over\sqrt{t}})+\theta^* ({x\over\sqrt{t}})+\psi(x,t) \en As for the $s$ variable, one gets \be s(x,t)=s^*(x,t)+r^*(x,t)+v(x,t) \en where \be s^*(x,t)=s^*({x\over\sqrt{t}}),\;\; r^*(x,t)={1\over\sqrt{t}}r^*({x\over\sqrt{t}}) \en are fixed points ``slaved'' respectively to $\phi^*$ and $\theta^*$. They satisfy the boundary conditions \be &&\lim_{x\rightarrow\pm \infty}|s^*(x)-s_\pm |=0\; ,\; \lim_{x\rightarrow\pm \infty}r^*(x)=0\; .\; \en The equations for $\psi$ and $v$ are rather complicated, but are essentially of the form heat equation plus irrelevant terms, in the sense of the previous section. Since now $\psi$ and $v$ decay to zero at infinity, we are in the situation of that section and, taking small initial data (in a suitable norm), one shows that the corresponding solution diffuses to zero. This, then, allows us to prove equation (4). \subsection{Stability of Fronts in the Ginzburg-Landau Equation.} Let us write the Ginzburg-Landau equation (1) in radial and angle variables, $ u= re^{i\phi}$: \be r_t =r_{xx} + r (1 - \phi^2_x) - r^3 \en \be \phi_t = \phi_{xx} + 2 r^{-1} \phi_x r_x \en and let us discuss the stability of the front solutions of these equations. It is well known that these equations have real, positive, front solutions, i.e. solutions of the form \be \phi=0, r=r_c (x-ct) \geq 0, \en such that $r_c$ interpolates between a stable and an unstable solution of (32), i.e. $r_c \rightarrow +1$ for $x \rightarrow - \infty, r_c \rightarrow 0$, for $x \rightarrow + \infty$. Indeed, from (32,34), we see that $r_c$ satisfies \be r_c^{''} + c r'_c + r_c - r^3_c = 0 \en which, if we reinterpret the variable as ``time", can be seen as Newton's equation of motion of a particle of mass one subjected to a friction term $c r'_c$ and to a force deriving from the potential $\frac{r^2}{2} - \frac{r^4}{4}$, which is an inverted double-well. It is intuitively clear and easily proved that, for $c$ not too small, solutions exist that satisfy the required conditions, i.e. such that $r_c$ tends, as ``time" goes to $+ \infty$, to zero, the stable critical point of the potential, and to one as ``time" goes to $- \infty$. For large ``time" $u=x-ct,r_c (u)$ will decay exponentially, as is seen from the linearization of (35) at $r=0$. One gets \be r_c (u) \leq (C_1+C_2u) e^{- \gamma u} \en where $\gamma$ is given by $\gamma^2 - c\gamma + 1 = 0 $ i.e. \be \gamma_c = \ha(c - \sqrt{c^2 -4}) \en which is real for $c \geq 2$, in which case $ \gamma_c \leq 1$ (actually, one can take $C_2=0$ in (36), if $\gamma <1$). Thus, the larger the friction, the slower the decay. For $c <2$, the solution ``overshoots" the minimum at zero, i.e. $r_c$ is no longer positive. Each of the solutions $r_{c}$ with $c \geq 2$ is stable under real perturbations $(\phi = 0):$ if we start with initial data $r(x,0)$, with $r=r_{c} + s $ with $0 \leq r \leq 1$, $s$ decaying faster than $e^{- \gamma_c x}$ for $x \rightarrow + \infty$, $r(x,t)$ will converge, as $t \rightarrow + \infty$, to $r_{c} (x-ct)$, see \cite{AW,Br,CE1}. However the solution with $c=2, \gamma_c =1$ is more stable than the others in the sense that any initial data $r (x,0)$ with $0 \leq r \leq 1$ which decays faster than $e^{-x}$ as $x \rightarrow + \infty$ (in particular, if $r$ is of compact support) will converge, as $t \rightarrow + \infty$, to $r_{2} (x-2t)$ \cite{AW,Br}. From now on, we shall concentrate on the most interesting front, namely the one with $c=2, \gamma = 1$ and we write $r$ for $r_{2}$ . We consider a complex perturbation of $r : r(x,0) = r + s$ with $\phi (x,0) \neq 0$ and $\phi (x,0), s(x,0)$ are small in a suitable sense. The equations satisfied by $\phi$ and $s$ are : \be \phi_t = \phi_{xx} + 2(r_x \phi_x + s_x \phi_x) (r + s)^{-1} \en \be s_t = s_{xx} + s (1-3 r^2) - r \phi^2_x - s \phi^2_x -3 r s^2 - s^3 \en Since $r$ is a function of $x-2t$ it is convenient to consider also the equation in the frame of reference of the front : let $u=x-2t$ and $\phi_f (u,t) = \phi (u+2t,t)$,$s_f (u,t) = s (u+2t,t)$; then $\phi_f$ and $s_f$ satisfy equations like (38,39), with $2 \phi_{fu}$ added to the RHS of (38) and $2 s_{fu}$ added to the one of (39). Now $r = r (u)$ is time-independent. To understand the expected behaviour of $\phi_f (u,t)$, let us consider the linearized equation around the zero solution : \be \phi_{ft} = \phi_{fuu} + 2 r_u \phi_{fu} r^{-1} + 2 \phi_{fu}. \en It is convenient to rewrite this equation as a heat equation with a potential: Let \be \phi_f (u,t) = e^{-u}r(u)^{-1} \psi (u,t). \en Then, $\psi$ satisfies \be \psi_t = \psi_{uu} - V\psi \en with \be V = 1 + \frac{r''}{r} + 2 \frac{r'}{r}= r^2. \en To derive the last equality, we used eq.(35), satisfied by $r$. Since $r \simeq 1$ for $u \rightarrow - \infty$, $r \simeq e^{- u}$ for $u \rightarrow + \infty$, we have $V \simeq 1$ for $u \rightarrow - \infty, V \simeq 0$ for $u \rightarrow + \infty$. So, starting with $\psi (u,0)$ localised around $u=0$, we expect that \be \psi (u,t) \sim \left \{ \begin{array}{ll} \frac{ e^{ -t-\frac{u^2}{4t}}}{\sqrt {t}} & u \rightarrow - \infty \\ \frac{ e^{ -\frac{u^2}{4t} }}{\sqrt t} & u \rightarrow + \infty. \end{array} \right. \en Hence, using (41) and the asymptotic behaviour of $r(u)$, \be \phi_f (u,t) \sim \left \{ \begin{array}{ll} \frac{e^{- t-\frac{u^2}{4t}- u}}{\sqrt t} & u \rightarrow - \infty \\ \frac{ e^{-\frac{u^2}{4t}}}{\sqrt t} & u \rightarrow + \infty \end{array} \right. \en which can be written as \be \phi_f (u,t) \sim \left \{ \begin{array}{ll} \frac{ e^{ - \frac{(u + 2t)^2}{4t} } }{\sqrt t} & u \rightarrow - \infty \\ \frac{ e^{ - \frac{u^2}{4t}}}{\sqrt t} & u \rightarrow + \infty. \end{array} \right. \en Since $u+2t=x$, the first part of (46) is a diffusive wave stationary in the fixed frame. The second part represents a diffusive wave which is ``carried along" by the front. This is a rough, but basically correct picture. Let us consider the linear equation for $s_f$, in the front frame : \be s_{ft} = s_{fuu} + s_{fu} + s_f (1-3 r^2) \en Writing $s_f=e^{-u} \sigma$, we get \be \sigma_t = \sigma_{uu} - \tilde{V} \sigma \en with $\tilde{V} = 3 r^2$. Following the analysis leading to (44), we get \be s(u,t) \sim \left \{ \begin{array}{ll} \frac{ e^{-3t-\frac{u^2}{4t}-u} } {\sqrt t} = \frac{ e^{-2t}}{\sqrt t} e^{-\frac{ (u+2t)^2}{4t}} & \mbox{as} \;\;\; u \rightarrow - \infty \\ \frac{ e^{-\frac{u^2}{4t}}} {\sqrt t} e^{-u} & \mbox{as} \;\;\; u \rightarrow + \infty \end{array} \right. \en There is a ``wave" which is stationary in the front frame, but exponentially decreasing in $u$ , while the wave which stays in the fixed frame is suppressed by the factor $e^{-2t}$. The rigorous results that justify this picture are of two types: Using the RG method, Gallay \cite{Ga} was able to obtain very precise asymptotics on how a small perturbation of {\it real fronts} diffuses to zero (this improves previous results of \cite{Ki,Sa}). So Gallay considers equation (39) with $\phi = 0$, in the front frame, whose linearization is given by (47). He writes \be s(u,t) = r' (u) w(u,t) \en and studies the behaviour of $w(u,t)$ as $t \to \infty$. Since $r(u)$ goes exponentially to 1 or 0 as $u$ goes to $-\infty$ or $+\infty$, $r'(u)$ will be localized around $u=0$. The main result is that $w$ has the following universal asymptotics \cite{Gal}: let $u=\xi \sqrt{t}$, then \be w(\xi \sqrt{t},t) \simeq A t^{-3/2} f^\ast (\xi) \en where $$ f^\ast(\xi) = \left\{ \begin{array}{lllll} 1 & \mbox{if} \; \xi \leq 0 \\ e^{-\xi^2/4} & \mbox{if} \; \xi>0 \end{array}\right. $$ and $A$ depends again on the initial conditions. The limit (51) holds in a weighted $L^1 \cap L^\infty$ norm. Going back to (47-49), the power $t^{-3/2}$ is easy to grasp intuitively: the potential $\tilde V$ in (48) plays the role of a barrier around $u=0$ ($\tilde V$ goes exponentially to 3 for $u \to - \infty$ and exponentially to $0$ far $u \to + \infty$). The simplest approximation is to replace the RHS of (48) by a Laplacian on ${\bf R}^+$ with Dirichlet boundary conditions at $u=0$. And that operator leads to a $t^{-3/2}$ decay of the solution. This effect was not taken into account in (49). The second type of results deals with the complex perturbations. In \cite{BK3}, we consider initial data in the Banach space of $C^1$-functions $\phi,s$ with the norm \be \|(\phi,s)\|=\sup_x(1+\mid x \mid)^{3+\delta}( \mid \phi (x) \mid + \mid \phi' (x) \mid+(1+e^x)( \mid s (x) \mid + \mid s' (x) \mid)) \en and prove \vs{3mm} \no {\bf Theorem 3}. {\it For any $\delta > 0$ there exists an $\epsilon > 0$ such that equations $(38,39)$ with initial data $\phi (x,1) = \phi (x), s(x,1) = s(x)$, and $\|(\phi,s)\|<\epsilon$, have a unique classical solution $\phi (x,t), s(x,t)$, for all $t \geq 1$, such that} \BE | \phi (x,t) | & \leq & t^{-\frac{1}{2} + \delta} \\ (1+e^{u}) |s(x,t)| & \leq & t^{-1+\delta} \EN \vs{2mm} \noindent {\bf Remark 1.} The power laws of the decay in time are presumably optimal (except for the $\delta$) and are different from those of Gallay, because the diffusive wave that is stationary in the fixed frame, for the $\phi$ variable (see (45)), goes only diffusively to zero. This in turn slows downs the decay of the $s$ variable, due to the nonlinear term $s \phi^2_x$ in (39). \noindent {\bf Remark 2.} The nonlinear terms in (38,39) turn out to be irrelevant, in the RG sense. However, this is not a simple affair: to show it, one has to take into account the precise decay of $\phi$ and $s$ both in the fixed and the front frames. This makes the proofs rather complicated. \noindent {\bf Remark 3.} Finally, let us mention that Eckmann and Wayne \cite{EW}, using a completely different (and simpler) method, namely coercive functionals, have proven similar results: they can consider a larger space of perturbations $(s,\phi)$ than the one defined by (52), but they do not obtain explicit upper bounds on the decay in time. \section{Universality in Blow-Up for Nonlinear Heat Equations} \setcounter{equation}{0} \subsection{Statement of the problem.} Let us now consider equations for which global existence results do not hold: the solutions of the initial value problem \be u_t = u_{xx} + u^p \en where $p > 1, u=u(x,t),x \in {\bf R}$, and $u(\cdot,0)=u_0\in C^0({\bf R})$, will, for a large class of initial data $u_0$, diverge in a finite time at a single point (for reviews on this problem, see \cite{HV,Le}). Again, we limit ourselves to one space dimension, but the generalization is straightforward. The RG ideas can be applied to the analysis of the profile of the solution at the time of blow-up. To explain what this means, let us fix the blow-up point to be 0 and the blow-up time to be $T$. Then, we ask whether it is possible to find a function $f^*(x)$ and a rescaling $g(t,T)$ so that \be \lim_{t \uparrow T} (T-t)^{\frac{1}{p-1}} \; u(g(t,T)x,t)=f^*(x) \en Moreover, we want to see how $g$ or $f^*$ depend on the initial data. The prefactor $(T-t)^{\frac{1}{p-1}}$ in (2) can be understood easily: for initial data $u_0(x)$ constant in $x$, $u(t)$ solves the ODE $ u_t = u^p$, i.e. $u(t) = ((p-1)(T-t))^{\frac{1}{1-p}}$ for $T=(p-1)^{-1}u_0^{1-p}$. In \cite{HV2,HV3,HV1,V} (see also \cite{F1,F2}) several possible $f^*$'s are discussed, and the set of initial data that will lead to a given $f^*$ is partially characterized. In \cite{BK4}, we showed that there exists, in the space of initial data $C^0({\bf R})$, sets ${\cal M}_k$ of codimension $2k$, such that, for $u_0 \in {\cal M}_k$, the limiting behaviour (2) is obtained, in the case $k=1$, for \BE g(t,T)&=&((T-t) | \log (T-t)|)^{\frac{1}{2}} \\ f^*(x)&=&\left(p-1+ b^*x^2\right)^{\frac{1}{1-p}} \EN where $b^*= \frac{(p-1)^2}{4p}$, and in the case $k>1$ for \BE g(t,T)&=&(T-t)^{\frac{1}{2k}}\\ f_b^*(x)&=&\left(p-1+ bx^{2k} \right)^{\frac{1}{1-p}}. \EN where now $b$ is an arbitary positive number. As shown in \cite{BK4}, one can also add suitable (irrelevant) terms to (1) without affecting the result. It was shown in \cite{HV2,HV1,V,F2} that, under quite general hypotheses, (3,4) or (5,6) are the only possibilities (see also \cite{VGH} for a formal analysis). Moreover, solutions that behave as in (5,6) for $k=2$ were constructed in \cite{HV2}. The codimension of ${\cal M}_k$ for $k=1$ is easy to understand: since we have fixed the blow-up point (to zero) and the blow-up time (to $T$), we have to fix two parameters in the initial data. To reach the other profiles, $2k-2$ additional parameters need to be fixed in the initial data. The $k=1$ situation is therefore the most generic one. In the RG language, $f^* $ and $f^*_b$ can be viewed as fixed points of a renormalization group transformation having $2k$ unstable (``relevant", in renormalization group terminology) directions. Thus, to converge towards the fixed point, one has to fine-tune $2k$ parameters (one for each unstable direction) and this explains why ${\cal M}_k$ is of codimension $2k$, and in what sense $f^*$, $f^*_b$ are ``universal". In addition, we encounter also one neutral (``marginal") mode, which, for $k=1$, turns out to be stable when nonlinear effects are taken into account and for $k>1$ parametrizes a curve of fixed points. Our results are perturbative, i.e. the sets ${\cal M}_k$ consist of initial data that are close to the corresponding fixed point. Therefore, our results are similar to those of Bressan \cite{Br} who considers a nonlinearity $e^{u}$ instead of $u^p$ and obtains the universal profile analogous to our $k=1$ case. The connection of blow-up and center manifold theory was used earlier in the work of Filippas, Kohn and Liu \cite{F1,F2} and of Herrero, Velazquez \cite{HV2,HV3,HV1,HV,V} and Galaktionov \cite{VGH}. Futhermore, the scaling and the dynamical systems aspect of our work goes back to Giga and Kohn \cite{GK,Kohn1,Kohn2}. Rescalings as in (7) below were used as a technique for numerical computation in \cite{Kohn3}. Let us first describe a change of variables that transforms the problem (1) into a problem of long time asymptotics: we write (1) in the ``blow--up--variables'': given a $u: {\bf R} \times [0,T) \to {\bf R}$, define $\phi : {\bf R} \times [-\log\ T,\infty) \to {\bf R}$ by \be u(x,t) = (T - t)^{-{1\over p-1}}\phi ({x\over (T - t)^{1/2k}}, -\log\ (T - t)). \en Then $u$ is a classical solution of (1) if and only if $\phi (\xi,\tau)$ is a classical solution of \BE \phi_\tau& =& L_\tau^{-2} \phi_{\xi\xi} - {1\over 2k}\xi \phi_\xi - {1\over p - 1}\phi + \phi^p \\ \phi (\xi,\tau_0)& =& T^{1\over p-1}u_0(T^{1\over 2k}\xi) \EN where $\tau_0=-\log T$, and \be L_{\tau} = e^{{1\over 2}\tau (1-1/k)}. \en We construct in \cite{BK4} global solutions of (8), with suitable initial data, thereby establishing blow--up for (1). Note that, for $k=1$, the scaling in (7) differs from the one used in (3) by a factor $| \log (T-t) |^{1/2}=\tau^{1/2}$. Actually, the situation where $k>1$ is easier to understand heuristically, so let us start by discussing the latter. \subsection{Analysis of $k>1$.} For $k>1$, as $\tau\to\infty$, the factor $L_{\tau}^{-2}$ in front of the second derivative in (8) leads us to consider the solutions of \be \phi_\tau = -{1\over 2k}\xi \phi_\xi - {1\over p-1} \phi + \phi^p\en Observe that the ''fixed points'' $f_b^*(x)=\left(p-1+ bx^{2k} \right)^{\frac{1}{1-p}}$ of (6) are stationary solutions of that equation. The latter can of course be integrated in closed form, but before doing that, let us first look at its linearization around the constant solution $\phi= (p-1)^{1\over 1-p}$. The linear problem is $\phi_\tau= {\cal L}_\infty\phi$, where \be {\cal L}_\infty = - {1\over 2k}\xi \frac{d}{d\xi} + 1 \; . \en and so, in the space of polynomials, we have now $2k$ expanding directions corresponding to $\xi^n$, for $n<2k$. Equation (11) is solved by putting $\phi (\xi,\tau) = e^{-{\tau\over p-1}} h(e^{-\tau/2k}\xi,\tau)$ whereby $\frac{d}{d\tau} h(y,\tau) = e^{-\tau}h(y,\tau)^p$ and so, for $\rho = \tau - \tau_0$ \be \phi (\xi,\tau) = {e^{-{\rho\over p-1}} f(e^{-\rho/2k}\xi)\over [1 - (p-1)f(e^{-\rho/2k}\xi)^{p-1}(1 - e^{-\rho})]^{1/p-1}} \en where $\phi (\xi,\tau_0) = f(\xi)$. The stationary solutions $f_b^*$ are stable in a suitable codimension $2k$ space: let us consider $f$ smooth, with \be f(0) = (p - 1)^{-{1\over p-1}},\ f^{(\ell)}(0) = 0\ \ \ell < 2k,\ f^{(2k)}(0) = \beta < 0\en and \be 0\leq f(\xi) < (p - 1)^{-{1\over p-1}} \ \;\; \xi \neq 0.\en Then, for all $\xi\in{\bf R}$, \be |\phi (\xi,\tau) - f^*_b(\xi)| \mathrel{\mathop{\longrightarrow}\limits_{\tau \to \infty}} 0\en where \be f_b^*(\xi) = (p - 1 + b\xi^{2k})^{-{1\over p-1}}\en for some $b$ depending on $\beta$,$ k$, $p$. These considerations thus lead us to expect (8) to have global solutions with initial data in a suitable codimension $2k$ set in a ball around (17) in a suitable Banach space. Of course, the perturbation $L^{-2}_{\tau}\phi_{\xi\xi}$ in (8) is very singular, but, basically, the picture is not much modified: the unstable modes turn out to be $\tau$-dependent Hermite functions instead of the monomials $\xi^n$, and one has to fine-tune the projection of the initial data on these Hermite functions instead of imposing the vanishing of the derivatives as in (14). See \cite{BK4} for details. \subsection{A non-conventional center manifold problem: $k=1$.} Consider now the case $k=1$. There are several differences with respect to the previous one: there is no damping factor in front of the second derivative in (8), and the asymptotics is given by (4), with a ''universal'' $b^*$, and an extra logarithmic factor in the definition (3) of $g(t,T)$. To understand the dynamics of (8) in that case, let us start by considering again its linearization around the constant solution $\phi= (p-1)^{1\over 1-p}$. The linear problem is $ \phi_\tau= {\cal L}\phi$, where now \be {\cal L} = \frac{d^2}{d\xi^2} - {1\over 2}\xi \frac{d}{d\xi} + 1 \; . \en Hence, the first thing we have to do, in order to understand the stability of the constant solution, is to study the spectrum of the linear operator ${\cal L}$. ${\cal L}$ is self--adjoint on ${\cal D}({\cal L}) \subset L^2({\bf R},d\mu)$ with \be d\mu (\xi) = \frac{e^{-\xi^2/4}d\xi}{\sqrt{4 \pi}} \en The spectrum of ${\cal L}$ is \be spec({\cal L}) = \{1 - {n\over 2}\mid n \in {\bf N}\} \en and we take as eigenfunctions multiples of Hermite polynomials \be h_n(\xi) = \sum\limits_{m=0}^{[{n\over 2}]}{n!\over m!(n-2m)!}(-1)^m {\xi}^{n-2m} \en that satisfy \be \int h_nh_md\mu = 2^n n!\delta_{nm} \en and \be {\cal L}h_n = (1 - {n\over 2})h_n. \en Thus the linearization of (8) at the constant solution, for $k=1$, has two expanding (``relevant'') modes, $h_0$ and $h_1$, and one neutral (``marginal'') one, $h_{2}=\xi^2-2$. Our goal is therefore to construct a center manifold for the flow of (8), in a neighbourhood of the fixed point. Formally, we would expand, \be \phi (\xi,\tau) = (p-1)^{\frac{1}{p-1}} + \psi (\xi,\tau) \;\; \mbox{as} \;\; \psi = \sum^\infty_{n=0} \psi_n (\tau) h_n (\xi), \en and rewrite (8) for the $\psi_n(\tau)$ as an infinite set of ODE's: $$\frac{d}{d\tau} \psi_n (\tau) = (1-\frac{n}{2}) \psi_n (\tau) + \; \mbox{nonlinear terms} $$ A formal solution of this flow yields (see below for the calculation): $\psi_2 (\tau) \simeq C_p (\log \tau) \tau^{-1}$ with $C_p = \frac{-(p-1)^{\frac{1}{1-p}}}{4p}$ as in (4). However, there are severe problems with this approach. Since the eigenfunction $h_n$ of the linearization $\cal L$ {\it increase} at infinity, the expansion (24) is not useful for $\xi$ large, and, in particular, we cannot use any standard infinite dimensional center manifold theorem. The key to the solution to this problem comes again from a scaling argument, which will explain the emergence of the fixed point $f^*$. Let \be \phi_L(\xi,\tau) = \phi (L\xi,L^{2}\tau) \en Then, $\phi_L$ satisfies the equation $$ H(\phi_L) = L^{-2}(- {\phi}_{L\tau} + \phi_{L\xi\xi}), $$ where we defined \be H(\phi)= {1\over 2}\xi \phi_\xi + {1\over p-1}\phi - \phi^p\; . \en Hence, as $L \to \infty$, we expect the solutions of $H(\phi) = 0$ to be relevant. These are (like the stationary solutions of (11)) given by the one-parameter family $f_b^*$, given by (6), with $k=1$. Therefore, we have the following picture: instead of perturbing around the constant solution, introduce $ \phi_b(\xi,\tau) = (p-1+ b\xi^{2}/\tau)^{1\over {p-1}}$, and write \be \phi (\xi,\tau) = \phi_b(\xi,\tau) + \eta (\xi,\tau). \en A local (i.e. for $\xi$ small) center manifold analysis then fixes $b=b^*$, as in (4). For large $\xi$, namely ${\xi^2\over \tau} > {\cal O}(1)$, the linearization $\tilde{\cal L}$ of (8) around $\phi_b$ differs from ${\cal L}$, and, actually, in that region (see below) $\tilde{\cal L} \simeq {\cal L}-2$. By (20), the spectrum of ${\cal L}-2$ is entirely negative. Hence, the dynamics tends to {\it contract} ${\eta}$. We will now explain the calculation that yields $b^*$ in (4). For simplicity of notations, we shall consider only $p=2$, i.e. \be \phi_b(\xi,\tau) = (1+ b\xi^{2}/\tau)^{-1}.\en We get, using $H(\phi_b) = 0$, \BE \eta_\tau & =& \eta_{\xi\xi} - H(\phi_b + \eta) + H(\phi_b) + \phi_{b\xi\xi} - \phi_{b\tau} \non\\ &=& ({\cal L} + W)\eta + M(\eta) + \phi_{b\xi\xi} - \phi_{b\tau} \EN where we introduce \BE &&W = 2(\phi_b -1)\\ &&M(\eta) = (\phi_b + \eta)^2 - \phi_b^2 - 2\phi_b \eta. \EN The operator $\cal L$, given by (18), has two unstable modes. Note that, formally, (i.e., for $\xi$ of order one) $W$ is ${\cal O}(\tau^{-1}),\ M$ is nonlinear in $\eta$ and $\phi_{b\xi\xi} - {\phi}_{b\tau}$ is ${\cal O}(\tau^{-1})$. We want to construct a center manifold for (29), i.e.\ to see how to fix two parameters in the initial data of $\eta$, such that the flow of (29) drives $\eta$ to zero. A simple calculation will show that this can only be achieved through a suitable choice of $b$ in (28). To see this, let us simplify further and consider $\eta$ even in $\xi$, which will imply that we need to fix only one parameter. This example contains all the relevant features of the general case. It is convenient to write \be \eta_0 (\tau) = \frac{a}{\tau} \en and define $\psi$ by $$ \eta = \eta_0 + \psi. $$ Then $\psi$ satisfies the equation \be \psi_\tau = \tilde{\cal L} \psi + N(\psi) + \alpha \en with \BE \tilde{\cal L}&=& {\cal L} + V \nonumber\\ V& =& 2(\phi_b + \eta_0 -1)\\ N(\psi) & =& (\phi_b + \eta_0 + \psi)^2 - (\phi_b + \eta_0)^2 - 2(\phi_b + \eta_0)\psi \\ \alpha& = &\phi_{b\xi\xi} - \phi_{b\tau} + ({\cal L} + W)\eta_0 - \eta_{0 \tau} + M(\eta_0) \nonumber \\ & =& \phi_{b\xi\xi}-\phi_{b\tau} + \eta_0+ W \eta_0 - \eta_{0\tau} + M (\eta_0). \EN Let us decompose $\psi$ (which is even in $\eta$) as \be \psi = \psi_0(\tau) + \psi_2(\tau)h_2 + \psi^{\perp} \en where $\psi^{\perp}$ is orthogonal to $h_n,\ n \leq 2$ . Next we expand $V$ and $\alpha$ (for $\xi={\cal O}(1))$: \BE &&V = -{2b\xi^2\over \tau} + {2a\over \tau} + {\cal O} ({1\over \tau^2}) \\ &&\alpha =(a-2b)\tau^{-1} + (a+a^2+ (12b^2 -b - 2ab)\xi^2))\tau^{-2} + {\cal O}(\tau^{-3}). \EN Inserting (32), (37) in (33) and retaining only the leading terms in $1/\tau$ and $\psi_i,\ i = 0,2$, we get from $\psi_{i\tau} = (2^i i!)^{-1}(h_i,\psi_\tau)$ ($(\cdot,\cdot)$ is the scalar product of $L^2({\bf R}, d\mu)$ and we use (22)): \BE \psi_{0\tau} &=& \psi_0 + (a - 2b)\tau^{-1} + R_0\\ \psi_{2\tau} &=& \beta \tau^{-1} \psi_2 + (12b^2 -b - 2ab)\tau^{-2} + R_2 \EN where $R_0 = {\cal O}(\tau^{-2} + \tau^{-1}|\psi| + |\psi|^2), R_2 = {\cal O}(\tau^{-3} + \tau^{-1}|\psi_0| + \tau^{-2}|\psi_2| + |\psi|^2)$,and $\beta = 2a - {1\over 4}b(\xi^2h_2,h_2) = 2a - 20b$ (coming from the $V \psi$ term in (33)). We choose now $a$ so that the ${\cal O}(\tau^{-1})$ term in $\psi_{0 \tau}$ vanishes i.e. \be a = 2b \en and $b$ such that the ${\cal O}(\tau^{-2})$ term in $ {\psi}_{2 \tau}$ is zero: \be b = b^*=1/8. \en Note that this choice correspond to $b=b^*$ in (4) for $p=2$. Then $\beta=-2$ and our equations read \be \psi_{0\tau} = \psi_0 + R_0,\ \ \psi_{2 \tau} = -{2\over \tau}\psi_2 + R_2. \en Now, keeping in mind the presence of the $R_0$, $R_2$ terms, \be \psi_0 = {\cal O}(\tau^{-2}),\ \ \psi_2 = {\cal O} ((\log\ \tau)\tau^{-2}) \en would be consistent solutions. Of course, we need to show that the expanding variable $\psi_0$ will satisfy (45) by a suitable choice of $\psi_0(\tau_0)$. This is rather easy to do, using the fact that $\psi_0$ is expanding; in the general case (with more than one parameter to fix), we used a topological argument. In the rigorous proof, we set up a suitable Banach space for the function $\eta$, parametrized by the $\psi_i$'s for $\xi^2/\tau<{\cal O}(1)$, and a function $\eta_l$ for $\xi^2/\tau>{\cal O}(1)$. The function $\eta_l$ will contract under the action of $\tilde{\cal L}$: indeed, from (34,28,32), we see that the potential $V$ tends to $-2$ as $\xi^2/\tau$ (and $\tau$) $\rightarrow \infty$. \section{Open Problems} \setcounter{equation}{0} There are several patterns and fronts in dissipative equations, and their stability can probably be studied using RG methods. For example, the Cahn-Hillard equation \cite{CH}: \be u_t=\Delta (-\Delta u -u -u^3) \en is often used to study the phase separation in alloys and fluids. One would like to study the stability (and possibly the dynamics), in infinite volume, of interface solutions of that equation. Another major open problem consists in the extension of the RG method to hyperbolic equations. The latter have their own scaling laws, see \cite{Le,St}. A lot is known on the stability of soliton solutions of (generalized) KdV equations \cite{PW}, but the stability of localized solutions of other nonparabolic equations is quite open. Also, there are open problem concerning the blow-up of solutions, most notably in the nonlinear Schr\"odinger equation \cite{LPSS}. So far, we have only considered equations whose solutions have a rather simple asymptotic behaviour. Of course, it is well known that finite dimensional dynamical systems described by differential equations can have a chaotic asymptotic behaviour, i.e. depend sensitively on the initial data but have also some statistical regularity in the sense that the long time average along the orbits is described by an invariant (SRB) measure \cite{ER}. For certain classes of $F$ in (1.1), one would like to find natural invariant measures for the flow. A class of dynamical systems, modeling PDE's, is obtained by discretizing space and time and considering a recursion \be u(x,t+1) = F(x,u(\cdot,t)) \en i.e. $u(x,t+1)$, with $x$ being a site of a lattice, is determined by the values taken by $u$ at time $t$ (usually on the sites in a neighbourhood of $x$). For a suitable class of $F$'s such dynamical systems are called Coupled Map Lattices (CML) \cite{Ka,Ka2}. The study of the invariant measures, and their properties, even for these "toy models" is essentially a terra incognita. In \cite {BS,PS,BK6} various ''high-temperatures'' (weak coupling) results are obtained. Almost nothing is known about the ''low temperatures'' (strong coupling) side (see \cite{Bu,BC,MH}). One would like to find models for which the set of invariant measures changes as the coupling is varied; this phenomenon would then be interpreted as a kind of nonequilibrium phase transition. \vs{5mm} \no{\Large\bf Acknowledgments} \vs{3mm} We would like to thank T. 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