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Long-time behavior of solutions to nonlinear reaction diffusion equations involving \(L^{1}\) data. (English) Zbl 1242.35059

The authors investigate the existence of a global attractor for the reaction diffusion problem \[ \begin{cases} u_t - \Delta u + f(u) = g&\text{in}\;\Omega \times \mathbb{R}^{+},\\ u(x,0)=u_0(x) &\text{in}\;\Omega,\\ u(x,t) =0 &\text{on}\;\partial \Omega \times \mathbb{R}^{+},\end{cases}\tag{RD} \] where \(\Omega \subset \mathbb{R}^{N}\), \(N \geq 2\), is a smooth bounded domain, \(u_0, g \in L^{1}(\Omega)\) and \(f\) is a function of class \(C^{1}\) satisfying the following conditions: There exist \(p\geq{2}\) and positive constants \(l\), \(C_1\) and \(C_2\) such that for all \(s \in \mathbb{R}\), \(f'(s) \geq {-l}\), \(C_1 |s|^{p} - k \leq f(s) s \leq C_1 |s|^{p} + k\) and \(|f'(s)| \leq C_2 (1+|s|^{p-2})\).
It is shown that the semigroup \(\{S(t)\}_{t \geq 0}\) generated by this problem possesses a global attractor \(\mathcal{A}\) in \(L^{1}(\Omega)\) which is invariant, compact in \(L^{p-1}(\Omega) \cap W_{0}^{1,q}(\Omega)\) with \[ q < \max\{N/(N-1),(2p-2)/p\} \] and attracting every bounded subset of \(L^{1}(\Omega)\) in the norm of \(L^{r}(\Omega) \cap H_{0}^{1}(\Omega)\) with \(r \in [1,+\infty)\). The proof is done by a decomposition technique combined with a bootstrap argument to establish some regularity results on the solutions.
The decomposition scheme involves the existence and uniqueness of solutions for the original problem (RD) to obtain regularity results for \(w(x,t)=u(x,t)-v(x)\) which satisfies \[ \begin{cases} w_t - \Delta w =f(v)-f(v+w) &\text{in }\Omega \times \mathbb{R}^{+},\\ w(x,0)=u_0(x)-v(x)&\text{in }\Omega,\\ w(x,t) =0 &\text{on }\partial \Omega \times \mathbb{R}^{+},\end{cases} \] where \(v\) satisfies the elliptic equation \(-\Delta v + f(v)=g \text{ in }\Omega\) with homogeneous Dirichlet boundary condition and \(v \in W_{0}^{1,q}(\Omega)\) for \(q < \max\{(2p-2)/p,N/(N-1)\}\).

MSC:

35B41 Attractors
35B65 Smoothness and regularity of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
35K58 Semilinear parabolic equations
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