\documentstyle[12pt,amsfonts]{article} \begin{document} \title{Well-posedness of a Semilinear Heat \\Equation with Weak Initial Data} \author{Jiahong Wu \\School of Mathematics\\The Institute for Advanced Study \\Princeton, NJ 08540 } \date{} \maketitle \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{define}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{example}[thm]{Example} \newtheorem{lemma}[thm]{Lemma} \def\theequation{\thesection.\arabic{equation}} \begin{abstract} This paper mainly consists of two parts. In the first part the initial value problem (IVP)of the semilinear heat equation $$\partial_t u- \Delta u =|u|^{k-1}u, \quad \mbox{on \quad {\Bbb R}^n\times (0,\infty)},\quad k\ge 2$$ $$u(x,0)=u_0(x),\qquad \qquad x\in{\Bbb R}^n\qquad\qquad\quad$$ with initial data in $\dot{L}_{r,p}$ are studied. We prove the well-posedness when $$10) and$$ \|u_\lambda(\cdot,t)\|_{\dot{L}_{r,p}}=\lambda^{r-\left(\frac{n}{p}- \frac{2}{k-1}\right)}\|u(\cdot, \lambda^2 t)\|_{\dot{L}_{r,p}} $$But unfortunately, we don't know if dimensional analysis always works and how to give a generally applicable criterion to detect the indices. \vspace{.13in} The problem of well-posedness for semilinear heat equation has attracted the attention of many mathematicians, but not many results related to very weak initial data have been published. The results concerning the IVP (\ref{eq:1.1}),(\ref{eq:1.2}) in L^p setting obtained by Weissler and others are as follows: for p\ge \frac{n(k-1)}{2}, there is well-posedness (\cite{w1},\cite{w2}); for 1\le p<\frac{n(k-1)}{2}, non-unique solutions can be constructed \cite{hw}. By letting r=0, i.e., p=\frac{n(k-1)}{2}, our results reduce to those in the L^p theory. In \cite{bp} Baras and Pierre prove that the IVP (\ref{eq:1.1}),(\ref{eq:1.2}) has a solution if and only if the initial measure is not too much concentrated. Clearly Sobolev spaces of negative indices contain distributions. In \cite{ky} Kozono and Yamazaki consider the IVP (\ref{eq:1.1}),(\ref{eq:1.2}) with initial data in real interpolation space {\cal N}^{r}_{p,q,\infty} (r=\frac{n}{p}-\frac{2}{k-1}), which is slightly larger than \dot{L}_{r,p} (by noting that {\cal N}^{r}_{p,p,\infty} =\dot{B}^{r}_{p,\infty} \supseteq \dot{L}_{r,p}, where \dot{B}^{r}_{p,\infty} is a homogeneous Besov space, see e.g. \cite{bl} p.147). However they obtain existence under the assumption that k\le q\le p and n(k-1)<2p0) denotes the class of all functions u \in C([0,T],H^s)\cap C((0,T], H^r) that also satisfy the condition$$ \|u\|_{BC_s((0,T], H^r)}\equiv \sup_{t\in[0,T]}\|(1+|\xi|^2)^{\frac{s}{2}} (1+|\xi|^2 t)^{\frac{r-s}{2}}\hat{u}(t)(\xi)\|_{L^2}< \infty $$Here \hat{f} denotes the Fourier transform of f. \vspace{.13in} We prove that if u_0\in H^s with s satisfying \begin{equation}\label{eqk:1.3} -1< s,\quad \mbox{for n=1},\qquad \frac{n}{2}-20 the solution u\in BC_s((0,T],H^r) for any r\ge s. See Theorem \ref{thm:4.1} of Section \ref{sec:4} for a precise statement. Clearly this result is not covered by the well-posedness result in the first part. \vspace{.13in} As we explained before, the index s=\frac{n}{2}-2 for n\ge 2 is exactly the number from dimensional analysis. But for n=1, our method only allow us to prove well-posedness for s>-1 and fails to extend to s>-3/2. It would be desirable to show that s=-1 is actually sharp by providing a counter example. \vspace{.13in} By taking special values of s and u_0, the well-posedness in this part reduces to some of those previously obtained by other authors (\cite{bf},\cite{w1}). Letting s=0, our result (for n\le 4) reduces to the L^p theory of Weissler and others (\cite{w1},\cite{w2}). In \cite{bf} Brezis and Friedman prove for u_0=\delta(x) that the solution exists for 0\frac{n}{2}-2, which is slightly more regular than \delta(x) since \delta(x)\not \in H^{s} ({\Bbb R}^n) \quad\mbox{for} \quad s>\frac{n}{2}-2. The fact that for n=2, \delta(x)\not \in H^{-1}({\Bbb R}^2), but \delta(x)\in H^{-1-\epsilon}({\Bbb R}^2) for any \epsilon>0, combined with the their non-existence result implies that our well-posedness result in 2-D is actually sharp. \vspace{.12in} The well-posedness result in this part is again proved by the contraction mapping arguments and we only deal with the IVP (\ref{eqk:1.1}),(\ref{eqk:1.2}) for s\le 0. The proof for s>0 can be given in a similar (and actually easier) way. \vspace{.2in} It is a great pleasure to thank Professor Carlos Kenig for helpful suggestions concerning this work. This work is supported by NSF grant DMS 9304580 at IAS. \newpage \section{Well-posedness in \dot{L}_{r,p}}\label{sec:2} \setcounter{equation}{0} First we define the spaces of weighted continuous functions in time, which have been introduced by Kato, Ponce and others in solving the Navier-Stokes equations (\cite{k2},\cite{kp1},\cite{kp2}). \begin{define} Suppose T>0 and \alpha\ge 0 are real numbers. The spaces  C_{\alpha,s,q} and \dot{C}_{\alpha,s,q} are defined as$$ C_{\alpha,s,q} \equiv \{f \in C((0,T), \dot{L}_{s,q}), \quad \|f\|_{\alpha,s,q} <\infty\} $$where the norm is given by$$ \|f\|_{\alpha,s,q}=\sup \{t^\alpha \|f\|_{s,q}, \quad t\in (0,T)\} $$\dot{C}_{\alpha,s,q} is a subspace of C_{\alpha,s,q}:$$ \dot{C}_{\alpha,s,q}\equiv\{f\in C_{\alpha,s,q}, \quad \lim_{t\to 0} t^\alpha\|f\|_{s,q}= 0\} $$When \alpha=0, \bar{C}_{s,q} are used for BC([0,T),\dot{L}_{s,q}). \end{define} These spaces are important in uniqueness and local existence problems (\cite{k2},\cite{kp1},\cite{kp2}). f\in C_{\alpha,s,q} (resp. f\in \dot{C}_{\alpha,s,q}) implies that \|f(t)\|_{s,q}=O(t^{-\alpha}) (resp. o(t^{-\alpha})). \vspace{.14in} The main result of this section is the well-posedness theorem that states \begin{thm}\label{thm:2.1} Assume u_0 \in \dot{L}_{r,p} with p and r satisfying \begin{equation}\label{eq:p} 10, there exists a unique solution u={\frak U}(u_0) to the IVP (\ref{eq:1.1}), (\ref{eq:1.2}) such that \begin{equation}\label{eq:2.1} u\in Y_T\equiv (\cap_{p\le q<\infty}\bar{C}_{r,q})\cap (\cap_{p\le q<\infty} \cap_{s>r}\dot{C}_{(s-r)/2,s,q}) \end{equation} In particular, (\ref{eq:2.1}) implies that$$ u\in BC([0,T), \dot{L}_{r,p})\cap (\cap_{s>r} C((0,T), \dot{L}_{s,p}) $$Furthermore, the mapping$${\frak U}: \Lambda\longmapsto Y_T: \quad u_0\longmapsto u$$is Lipschitz in some neighborhood \Lambda of u_0. \end{thm} We make several remarks concerning this theorem. \vspace{.1in} \begin{rem} It is easy to see from our proof of this theorem that if \|u_0\|_{\dot{L}_{r,p}} is sufficiently small, we may take T=\infty. \end{rem} \vspace{.1in} \begin{rem} The homogeneous spaces \dot{L}_{s,q} can be replaced by inhomogeneous spaces L_{s,q} (i.e., spaces of Bessel potentials):$$ L_{s,q}\equiv \{ v: \|v\|_{s,q}\equiv\|(1-\Delta)^{s/2} u\|_{L^q} <\infty\} $$to obtain quite similar well-posedness results. \end{rem} \vspace{.2in} We prove Theorem \ref{thm:2.1} by the method of integral equation and contraction-mapping arguments. This method has been extensively used by Kato, Ponce and others to prove the well-posedness of the Navier-Stokes equations in various type of functional spaces (\cite{k1},\cite{k2},\cite{kf},\cite{kp1},\cite{kp2}). First we write Equation (\ref{eq:1.1}) in the integral form$$ u=U u_0(t)+G(|u|^{k-1}u)(t)\equiv e^{-\Delta t}u_0 +\int_{0}^{t} e^{-\Delta(t-\tau)}(|u|^{k-1}u)(\tau) d\tau $$Then we estimate the operators U and G separately. The main estimates are established in the propositions that follow. \begin{prop}\label{prop1} \begin{description} \item[(1)] If s\in {\Bbb R} and q\in [1,\infty), then$$ U u_0(t)\to u_0, \quad \mbox{ in $\dot{L}_{s,q}$\quad as $t\to 0$}. $$\item[(2)] If s_1\le s_2, q_1\le q_2, and$$ \alpha_2=\left(s_2-s_1 +\frac{n}{q_1}-\frac{n}{q_2}\right) $$then U maps continuously from \dot{L}_{s_1,q_1} into \dot{C}_{ \alpha_2,s_2,q_2}. \end{description} \end{prop} {\bf Proof.}\quad The proof of (1) involves the definitions of the norms and the dominated convergence theorem. See \cite{kp1},\cite{kf} for the proof of (2). \vspace{.2in} Now we give the estimates for the operator G:$$ G g(t)=\int_{0}^{t}e^{-\Delta(t-\tau)}g(\tau) d\tau $$\begin{prop}\label{prop2} If q_1,q_2, \alpha_1,\alpha_2, s_1 and s_2 satisfy$$ q_1\le q_2,  \alpha_1<1,\quad \alpha_2=\alpha_1-1 +\frac{1}{2}\left[ s_2-s_1+\frac{n}{q_1}-\frac{n}{q_2}\right]  0\le s_2-s_1<2-{n}\left(\frac{1}{q_1}-\frac{1}{q_2}\right) $$then G maps continuously from \dot{C}_{\alpha_1,s_1,q_1} to \dot{C}_{\alpha_2,s_2,q_2}. \end{prop} {\bf Proof.}\quad The proof of this proposition is quite similar to that of Lemma 2.3 in \cite{kp1}. We just want to point out that the restrictions$$ \alpha_1<1, \quad s_2-s_1<2-n\left(\frac{1}{q_1}-\frac{1}{q_2}\right) $$are imposed to guarantee the finiteness of a Beta function involved in the estimates of G. \vspace{.2in} Now we turn to the proof of Theorem \ref{thm:2.1}. \vspace{.1in} \noindent{\bf Proof of Theorem \ref{thm:2.1}.}\quad We consider two cases: r<0 and r=0. For r<0, we define$$ X=\bar{C}_{r,p}\cap \dot{C}_{-\frac{r}{2},0,p}, $$with norm for u\in X given by$$ \|u\|_X =\|u- U u_0\|_{0,r,p} +\|u\|_{-\frac{r}{2},0,p} $$and the complete metric space X_R to be the closed ball in X of radius R. Consider the operator {\cal A}(u,u_0):X_R\times \Lambda\longmapsto X$$ {\cal A}(u,u_0)(t)= U u_0(t) + G (|u|^{k-1}u)(t),\quad 0\le t0$is small enough and$\Lambda$chosen properly. To estimate$G$, we use Proposition \ref{prop2} with $$q_1=\frac{p}{k},\quad q_2=p,\quad \alpha_1=-\frac{k r}{2},\quad \alpha_2=\frac{l}{k},\quad s_1=0,\quad s_2=\frac{2l}{k}+r,$$ we obtain $$\|G (|u|^{k-1}u)\|_{\frac{l}{k},\frac{2l}{k}+r,p} \le c\||u|^{k-1} u\|_{-\frac{kr}{2}, 0,\frac{p}{k}} \le \|u\|_{-\frac{r}{2},0,p}^{k} \le c R^k$$ for all$l\in [0, -\frac{k^2}{2}r)$. Here it is important to notice that the restrictions on$p$and$r$(\ref{eq:p}), (\ref{eq:r}) are necessary in order to apply Proposition \ref{prop2}. \vspace{.12in} Furthermore, $$\|{\cal A}(u,u_0)-{\cal A}({\tilde{u},u_0})\|_X \le \|G (|u|^{k-1}u) - G (|\tilde{u}|^{k-1}\tilde{u})\|_X$$ $$\le \|G (|u-\tilde{ u}|^{k-1} u)\|_X + \|G (|u-\tilde{u}| |\tilde{u}|^{k-1})\|_X$$ Using Proposition \ref{prop2} again $$\|{\cal A}(u,u_0)-{\cal A}({\tilde{u},u_0})\|_X \le 2\||u-\tilde{u}|^{k-1}u\|_{-\frac{kr}{2}, 0, \frac{p}{k}} +2\||u-\tilde{u}||\tilde{u}|^{k-1}\|_{-\frac{kr}{2}, 0, \frac{p}{k}}$$ $$\le c\|u\|_X\|u-\tilde{u}\|_{X}^{k-1} +c\|u-\tilde{u}\|_X \|\tilde{u}\| _{X}^{k-1}$$ So if we choose$T$to be small and$R$properly, then${\cal A}(u,u_0)$maps$X_R$into itself and is a contraction map when$k\ge2 $. Consequently there exists a unique fixed point$u\in X_R$:$u={\frak U} (u_0)$satisfying$u={\cal A}(u,u_0)$. It is easy to see from the above estimates that the uniqueness can be extended to$X_{R'}$for all$R'$by reducing the time interval and thus to the whole$X$. \vspace{.1in} To show that$u$is in the class of$Y_T$, we notice $$u(t)={\cal A}(u,u_0)(t)\equiv U u_0(t) + G(|u|^{k-1} u)(t),\quad t\in [0,T).$$ For small$s$, the above formula, combined with Proposition \ref{prop2}, can be used to prove that$u\in Y_T$. For large$s$,$u\in Y_T$can be shown by induction (see an analogous argument in \cite{k2} (p.60)). \vspace{.10in} To prove the Lipschitz continuity of${\frak U}$, let$u={\frak U}(u_0)$and$v={\frak U}(v_0)$for$u_0, v_0\in \Lambda$. Then $$\|u-v\|_X =\|{\cal A}(u,u_0)-{\cal A}(v,v_0)\|_X$$ $$\le \|{\cal A}(u,u_0)-{\cal A}(v,u_0)\|_X +\|{\cal A}(v,u_0)- {\cal A}(v,v_0)\|_X$$ $$\le \gamma\|u-v\|_X +\|U (u_0-v_0)\|_X$$ For small$T$and properly chosen$\Lambda$,$\gamma<1$since the mapping is a contraction and we obtain asserted result by using Proposition \ref{prop1} to the second term. \vspace{.16in} In the case$r=0$, we define $$X=\bar{C}_{0,p} \cap \dot{C}_{\frac{1}{4(k-1)}, 0,\frac{4p}{3}}$$ with the norm $$\|u\|_X =\|u- U u_0\|_{0,0,p} + \|u\|_{\frac{1}{4(k-1)},0,\frac{4p}{3}}$$ and$X_R$is again the closed ball in$X$of radius$R$. By Proposition \ref{prop2}, $$\| G (|u|^{k-1}u)\|_X = \| G(|u|^{k-1}u)\|_{0,0,p} + \| G(|u|^{k-1}u)\|_{\frac{1}{4(k-1)},0,\frac{4p}{3}}$$ $$\le c \|u^k\|_{\frac{k}{4(k-1)},0,\frac{4p}{3k}}\le c\|u\|^{k}_{ \frac{1}{4(k-1)},0,\frac{4p}{3}} \le c R^k$$ and the rest of the proof reduces to the previous case. This completes the proof of Theorem \ref{thm:2.1}. \newpage \section{Non-uniqueness for$r<\frac{n}{p}-\frac{2}{k-1}$}\label{sec:3} \setcounter{equation}{0} In this section we consider the situation when $$1< p< \frac{n(k-1)}{2} 0, there exists at least one non-trivial solution \Phi to the IVP (\ref{eq:3.1}), (\ref{eq:3.2}) such that$$ \Phi\in C([0,T), \dot{L}_{r,p})\cap C((0,T), \dot{C}_{-r/2,0,p}) $$Thus we get at least three different solutions \Phi, -\Phi and 0, corresponding to the same initial data 0. \end{thm} \vspace{.2in} We seek solutions to Equation (\ref{eq:3.1}) of the self-similar form$$ \Phi(x,t)= t^{-\frac{1}{k-1}}\omega(\frac{x}{\sqrt{t}}) $$Then the Equation (\ref{eq:3.1}) which \Phi should satisfy reduces to an O.D.E. of \omega,$$ \Delta \omega(x) +\frac{x}{2}\cdot \nabla\omega(x) +\frac{\omega(x)}{k-1} +|\omega(x)|^{k-1} \omega(x) =0, \quad x\in {\Bbb R}^n $$By assuming \omega is radial, i.e., \omega(x)=v(|x|) with v: [0, \infty)\longmapsto {\Bbb R}, the equation is further reduced to \begin{equation}\label{eq:3.4} v''(x) +\left(\frac{n-1}{x} +\frac{x}{2}\right)v'(x) +\frac{v(x)}{k-1} +|v(x)|^{k-1}v(x)=0, \quad x>0 \end{equation} \vspace{.14in} Haraux and Weissler \cite{hw} consider the solutions of Equation (\ref{eq:3.4}) and we need to use the following result of theirs. \begin{prop}\label{prop:3.1} Let k>1 and n\ge 1. If$$ 1 <\frac{n(k-1)}{2}< k+1 $$then for some v_0, there is a unique solution v\in C^2([0, \infty)) to Equation (\ref{eq:3.4}) with v(0)=v_0 and v'(0)=0 such that$$ \lim_{x\to \infty}x^m v(x) =0,\quad \mbox{for all$m>0$} $$\end{prop} \vspace{.2in} To prove the theorem, we only need to prove the following assertions about the solution \Phi=t^{-\frac{1}{k-1}}v(\frac{|x|}{\sqrt{t}}) constructed above. \begin{prop} Assume that the indices k,n,p and r satisfy (\ref{eq:3.3}). Then \begin{description} \item[(1)]$$ \Phi(t)\to 0,\qquad \mbox{in}\quad S'({\Bbb R}^n)\quad\mbox{ as}\quad t\to 0 $$where S'({\Bbb R}^n) is the space of tempered distributions. \item[(2)]$$ \Phi(t)\to 0, \qquad\mbox{in}\quad \dot{L}_{r,p}\quad\mbox{as}\quad t\to 0 $$\end{description} \end{prop} {\bf Proof.}\quad To prove assertion (1), we calculate for any \phi\in S,$$ \lim_{t\to 0}\int \Phi(x,t)\phi(x)dx =\lim_{t\to 0}\left( t^{-\frac{1}{k-1}} \int v(\frac{|x|}{\sqrt{t}})\phi(x)dx\right)  \le \lim_{t\to 0}\left( t^{-\frac{1}{k-1}+\frac{n}{2p}}\|v\|_{L^p} \|\phi\|_{L^q}\right),\quad \mbox{with} \quad \frac{1}{p}+\frac{1}{q}=1 $$Since \|v\|_{L^p} is finite as implied by Proposition \ref{prop:3.1}, we conclude that the above limit is zero. \vspace{.14in} To prove (2), we need the following lemma. \begin{lemma}\label{lem:3} Let q\in (1,\infty), 00, there is a unique solution u(t) of the IVP (\ref{eqk:1.1}),(\ref{eqk:1.2}) on the time interval [0,T] satisfying$$ u\in BC_s((0,T],H^r),\quad \mbox{for any$r\ge 0$} $$Furthermore, for any T'\in (0,T), there exists a neighborhood V of u_0 in H^{s} such that the mapping$$ \Phi: V\longmapsto BC_s((0,T'], H^{r}), $$is Lipschitz. \end{thm} \begin{rem} The theorem remains unchanged if the nonlinear term u^2 in Equation (\ref{eqk:1.1}) is replaced by -u^2. At this point the nonlinear heat equation differs from the nonlinear Schr\"{o}dinger equation for which the focusing and defocusing cases are quite different (see e.g. Bourgain \cite{b}). \end{rem} The proof of this theorem is again based on the contraction mapping principle. We write Equation (\ref{eqk:1.1}) in the integral form$$ u=U (u_0)(t)+G(u^2)(t)\equiv e^{-\Delta t}u_0 +\int_{0}^{t} e^{-\Delta(t-\tau)}(u^2)(\tau) d\tau $$Then we estimate the operators U and G on BC_s((0,T],H^r). The main estimates are established in the propositions that follow. \vspace{.16in} \begin{prop} \label{prop21} Let 00, \Phi maps X_{T,R} into X_{T,R} and is a contraction. Thus there is a unique fixed point u=\Phi(u) in X_{T,R}. It is clear that by reducing the time interval (0,T), we can extend the uniqueness to X_{T,R'} for any R' and thus to the whole class in X_T. \vspace{.1in} For T'\in (0,T), we can see from (\ref{tk}) that K can be replaced by a larger K' such that (\ref{tk}) still holds for K' and T'. That is, \Phi is still a contraction map for v_0\in V where V is some neighborhood of u_0. The Lipschitz continuity of \Phi is easily obtained by using tha fact that \Phi is contraction map on V. This finishes the proof of this theorem. \vspace{.22in} Now we state a lemma which will be used in the proof of Proposition \ref{prop22}. In what follows we will denote (1+|\xi|^2)^{1/2} by w(\xi) where \xi \in {\Bbb R}^n. \begin{lemma} \label{lem} Let r\ge 0,a be real numbers. If g,h \in H^r, then$$ \|w(\xi a)^r\widehat{gh}(\xi)\|_{L^\infty}\le \|w(\xi a)^r\hat{g}(\xi)\|_{L^2}\|w(\xi a)^r\hat{h}(\xi)\|_{L^2} $$\end{lemma} The proof of this lemma is simple and can be found in \cite{dix1}. \vspace{.22in} \noindent{\bf Proof of Proposition \ref{prop22}}. \quad First we estimate \|u_G\|_{BC_s((0,T], H^q)}. We only need to prove for the case r\le q\le r+2-\frac{n}{2} since the norm is a nondecreasing function of q. It is easy to check that for 00, b>0 the Beta function$$ B(a,b)=\int_{0}^{1}(1-x)^{a-1} x^{b-1} dx $$is finite. II can be estimated in a quite similar way and the final result is the same as that of I apart from the constant C may be different. \vspace{.12in} Now we show that u_G: (0,T]\to H^q is continuous. Let t_1,t_2\in (0,T] and we estimate the difference$$ \|u_G(t_2)-u_G(t_1)\|_{H^q} \le\left\|w(\xi)^q\int_{t_1}^{t_2}e^{-|\xi|^2(t_2-\tau)} \widehat{u^2}(\tau)(\xi) d\tau\right\|_{L^2}  +\left\|w(\xi)^q\int_{0}^{t_1}\left[e^{-|\xi|^2(t_2-\tau)}-e^{-|\xi|^2 (t_1-\tau)}\right]\widehat{u^2}(\tau)(\xi)d\tau \right\|_{L^2} =III+IV $$In a similar manner III and IV can be estimated and consequently we can show that for s\le q\le r+2-\frac{n}{2}:$$ III,\quad IV \to 0,\quad \mbox{as} \quad t_2-t_1\quad \to 0 $$\vspace{.12in} Finally we show that$$ u_G(t)\to 0,\quad \mbox{in$H^s$as$t\to 0$} $$We estimate$$ \|u_G\|_{H^s}=\left\|w(\xi)^s\int_{0}^{t}e^{-|\xi|^2(t-\tau)}\widehat{u^2} (\tau)(\xi)d\tau\right\|_{L^2}$$and this can be done similarly as before. We omit details. \newpage \begin{thebibliography}{99} \bibitem{bp} P. Baras, M. Pierre, {\em Problems paraboliques semilineaires avec donnees measures}, Applicable Analysis, {\bf 18}(1984), 111-149. \bibitem{bl} J. Bergh, J. 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