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Zbl 1242.35059
Zhong, Chengkui; Niu, Weisheng
Long-time behavior of solutions to nonlinear reaction diffusion equations involving $L^{1}$ data. (English)
[J] Commun. Contemp. Math. 14, No. 1, 1250007, 19 p. (2012). ISSN 0219-1997

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 $|f'(s)| \leq C_2 (1+|s|^{p-2})$.\par 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. \par 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 $q < \max\{(2p-2)/p,N/(N-1)\}$.
[Jauber C. Oliveira (Florianopolis)]

MSC 2000:: *35B41 Attractors
35B65 Smoothness of solutions of PDE
35K20 Second order parabolic equations, boundary value problems
35K57 Reaction-diffusion equations
35K58

Keywords: reaction diffusion equations; global attractor; $L^{1}$ data; decomposition technique

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Zentralblatt MATH Berlin [Germany]