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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).

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
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