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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 $$\cases u_t - \Delta u + f(u) = g&\text{in}\;\Omega \times \mathbb{R}^{+},\\ u(x,0)=u_0(x) &\text{in}\;\Omega,\\ u(x,t) =0 &\text{on}\;\partial \Omega \times \mathbb{R}^{+},\endcases\tag{RD}$$ 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 $s \in \Bbb{R}$, $f'(s) \geq {-l}$, $C_1 |s|^{p} - k \leq f(s) s \leq C_1 |s|^{p} + k$ 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 $$\cases w_t - \Delta w =f(v)-f(v+w) &\text{in }\Omega \times \mathbb{R}^{+},\\ w(x,0)=u_0(x)-v(x)&\text{in }\Omega,\\ w(x,t) =0 &\text{on }\partial \Omega \times \mathbb{R}^{+},\endcases$$ 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 $q < \max\{(2p-2)/p,N/(N-1)\}$.

35B41Attractors (PDE)
35B65Smoothness and regularity of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
35K57Reaction-diffusion equations
35K58Semilinear parabolic equations
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