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On the long time behaviour of solutions to nonlinear diffusion equations on $$\mathbb{R}^ n$$. (English) Zbl 0872.35014
The author studies the behaviour for $$t\to\infty$$ of solutions $$u=u(x,t)$$ of the parabolic equation: $u_t=\Delta u- f(u),\;x\in\mathbb{R}^N,\;t>0,\;|u(x,t)|\to 0\text{ for }|x|\to\infty\text{ and }u(0)= u_0\in L^2(\mathbb{R}^N).\tag{1}$ The nonlinearity $$f\in C^1(\mathbb{R})$$ satisfies: (2) $$f'(z)>-c$$ for $$z\in\mathbb{R}$$ and some $$c>0$$, $$f(0)=0$$ and $$f'(0)=a>0$$. Strongly related to (1) is the stationary problem: (3) $$\Delta w=f(w)$$, $$w\in W^{1,2}(\mathbb{R}^N)$$.
Based on the notion of concentrated compactness, the author proves some highly interesting results about the asymptotic behaviour of solutions $$u(x,t)$$ of (1) as $$t\to\infty$$. The basic assumptions needed for such results are that $$|u(t)|_{L^2}$$ remains bounded as $$t\to\infty$$ or less restrictively: (4) there is a sequence $$t_n\to\infty$$ such that $$\sup_n|u(t_n)|_{L^2}<\infty$$.
We cite two of the four main results proved in the paper. The first is: (A) Let $$u(t)\in L^2(\mathbb{R}^N)$$, $$t>0$$ be a solution of (1) such that (4) holds for some sequence $$t_n$$; then there exist solutions $$w_j$$, $$j=1,\dots,k$$ of (3), not necessarily distinct, a subsequence of $$t_n$$ (identified with $$t_n$$) and points $$x^n_j$$, $$j\leq k$$, $$n<\infty$$ such that $|u(\cdot,t_n)- w_1- \sum^k_{j=2} w_j(\cdot- x^n_j)|_X\to 0,\quad t_n\to \infty$ ($$X= W^{1,2}(\mathbb{R}^N)\cap C^2(\mathbb{R}^N))$$, where $$|x^n_j|\to\infty$$, $$\text{dist}(x^n_j,x^n_i)\to\infty$$ for $$i\neq j$$ and $$n\to\infty$$; if $$u(\cdot,t_m)>0$$ for some $$m$$ then $$w_j>0$$ for $$j=2,\dots,k$$. The proof of (A) uses results of P. L. Lions on concentrated compactness which provide the equilibrium solutions $$w_j$$.
The last of the author’s results deals with the existence of soliton like solutions of (1). To this end, he considers the equation: $u_t=\Delta u-au+|u|^{\alpha-1}u,\tag{5}$ $$x\in\mathbb{R}^N$$, with $$u(0)= u_0$$ in $$L^2(\mathbb{R}^N)$$. Here $$1<\alpha<\alpha^*$$, where $$\alpha^*= N(N-2)^{-1}$$ if $$N>3$$ and $$\alpha^*=(N+ 4)N^{-1}$$ otherwise. The result then is: (B) There is $$u_0\in W^{1,2}(\mathbb{R}^N)$$ and a solution $$w>0$$ of (3) such that the solution $$u$$ of (5) with $$u(0)=u_0$$ satisfies $|u(\cdot,t)- \sum^k_{j=1} (w(\cdot- x^+_j(t))- w(\cdot- x^-_j(t)))|_X\to0\quad\text{as}\quad t\to\infty;$ here $$x^+_j(t)$$, $$t>0$$, $$j=1,\dots,k$$ are such that $$|x^+_j(t)|$$, $$\text{dist}(x^+_j(x),x^+_i(t))\to\infty$$ for $$i\neq j$$ and $$t\to\infty$$, and the $$x^-_j$$ are obtained from $$x^+_j$$ by reflection with respect to the hyperplane $$x_1=0$$. Either from (B) itself or else by inspection of its proof one reads off that $$\gamma\leq|u(t)|_{L^2}$$, $$|u(t)|_{W^{1,2}}\leq\delta$$ for $$t>0$$ and some $$\gamma,\delta>0$$. The existence of such soliton like solutions as asserted by (B) is of considerable interest and should be investigated for other types of equations besides (5).

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations
##### Keywords:
concentrated compactness; soliton like solutions
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