1$, there exists a solution $f^*_2$ which decays at infinity like $|x|^{-{2\over{p-1}}}$ \cite{Ga,KP1}. Note that $f^*_2$ is integrable only for $p<3$. Rather detailed results are known on the basin of attraction of the various fixed points; we state them loosely, see the references for more precise statements e.g. on the type of convergence; also, in each case, the decay in time of $u(x,t)$ is $t^{-\alpha\over 2}$, as in (2.10), where the exponent $\alpha$ is related to the fixed point as in (2.3, 2.4). 1) For $p\geq 3$, and initial data $u(x,1)$ non-negative and integrable, the asymptotic behaviour of the solution is governed by $f^*_0$, as in Theorems 1, 2, with logarithmic corrections for $p=3$ \cite{Ga,GV}. This is a global result ($u(x,1)$ is not assumed to be small); our results are perturbative, and restricted to integer $p$ (but a general non-linearity $F$), but do not require $u(x,1)$ to be pointwise positive and hold (when $p>3$) also if one has $+u^p$ in (1); in that case, smallness of $u(x,1)$ is necessary since large initial data blow up in a finite time \cite{F,CEE,L}. 2) For $p<3$ and $u(x,1)$ non-negative and having (suitable) Gaussian decay, the asymptotic behaviour is governed by the non-trivial fixed point $f^*_1$ \cite{Ga,KP2}. 3) If one starts with $u(x,1)$ non-integrable and decaying at infinity like $|x|^{-\alpha}$, with $0<\alpha <1$, then, for $p>3$, the relevant "Gaussian" fixed point is $f^*_{\alpha}$ where ${\hat f^*_{\alpha}}(k) =|k|^{\alpha-1}e^{-k^2}$, which, for any $\alpha$, is a fixed point of (2.4), where $u$ solves the heat equation. This fixed point has the right $|x|^{-\alpha}$ decay at infinity. Now $u^p$ is relevant, marginal, or irrelevant according to whether $p< 1+{2\over \alpha}$, $p= 1+{2\over \alpha}$, or $p> 1+{2\over \alpha}$. One knows \cite{KP1,GV} that, for a non-negative initial data, the solution converges to $f^*_{\alpha}$ for $p> 1+{2\over \alpha}$. For $p= 1+{2\over \alpha}$, it converges to $f^*_2$. For $p< 1+{2\over \alpha}$, the solution converges to a solution constant in space, $(p-1)^{-{1\over p-1}}t^{-{1\over p-1}}$, which solves (1) without the diffusive term: $\dot{u}= - u^{p}$, and which can be viewed as a (somewhat degenerate) new fixed point. To see the analogy with the theory of critical phenomena, consider an Ising model or a $\phi^4$ theory, on an $N$ dimensional lattice, at the critical point. $N>4$ is like $p>3$ here: $\phi^4$ is irrelevant and the behaviour at the critical point is governed by the Gaussian fixed point. For $N=4$, $\phi^4$ becomes marginal, and the Gaussian behaviour is modified, like here in Theorem 2, by logarithmic corrections. However, this is not true for every marginal perturbation. In $\phi ^4$ theory, like here for $p=3$, this happens because the marginal term becomes irrelevant when higher order terms are included: $A_n$ and, therefore, $f_n$ go to zero which is the same thing as having a coupling constant in front of the $u^3$ term going to zero. In particular, this higher order irrelevancy depends, like in $\phi^4$ theory, on the sign of the perturbation. For $+u^3$ in (1), the solution blows up \cite{CEE,L}. In point 3) above, the marginal perturbation ($p= 1+{2\over \alpha}$), leads to a non-trivial fixed point, $f^*_2$ instead of $f^*_{\alpha}$. Also, in the Burgers' equation (3.36), the marginal term remains marginal to all orders and the solution is governed by a new fixed point, which is however easy to write down (see the end of Sect.4). For $N<4$, $\phi^4$ becomes relevant and one expects the critical behaviour to be governed by a non-trivial fixed point, whose existence is however much harder to establish than here. Another analogy with field theory concerns the constant $A$ in (7): this is like a "renormalised" constant whose corresponding "bare" value is $A_0 ={\hat f}(0)$. One of the problems encountered in proving (7), instead of just a bound on $u(x,t)$, is that $A$, unlike $A_0$, may depend in a complicated way on $f$ and $F$, and is not known explicitely. This usually limits the power of ordinary perturbation theory: one may try to expand $u(x,t)$ around $A_0 t^{-\ha} f^*_0$ which has the wrong constant and the perturbation series may not converge. One of the advantages of the RG method is that it allows to "build up" $A$ through a convergent sequence of approximations, as we did in the proof. Also, note that in the marginal case, the renormalisation of $A$ drives it to zero and produces the logarithmic correction in (39). Finally, one may also interpret in RG language the results on the existence of singular or very singular solutions of (1). A solution is $singular$ if, as $t\rightarrow 0$, $u(x,t)$ becomes concentrated on a point, and it is $very$ $ singular$ if, moreover, $\int dx u(x,t)$ diverges when $t\rightarrow 0$. For the family of equations (1), one knows that, for $1