# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

$\left\{\begin{array}{cc}{u}_{t}-{\Delta }u+f\left(u\right)=g\hfill & \text{in}\phantom{\rule{0.277778em}{0ex}}{\Omega }×{ℝ}^{+},\hfill \\ u\left(x,0\right)={u}_{0}\left(x\right)\hfill & \text{in}\phantom{\rule{0.277778em}{0ex}}{\Omega },\hfill \\ u\left(x,t\right)=0\hfill & \text{on}\phantom{\rule{0.277778em}{0ex}}\partial {\Omega }×{ℝ}^{+},\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(\mathrm{RD}\right)$

where ${\Omega }\subset {ℝ}^{N}$, $N\ge 2$, is a smooth bounded domain, ${u}_{0},g\in {L}^{1}\left({\Omega }\right)$ and $f$ is a function of class ${C}^{1}$ satisfying the following conditions: There exist $p\ge 2$ and positive constants $l$, ${C}_{1}$ and ${C}_{2}$ such that for all $s\in ℝ$, ${f}^{\text{'}}\left(s\right)\ge -l$, ${C}_{1}{|s|}^{p}-k\le f\left(s\right)s\le {C}_{1}{|s|}^{p}+k$ and $|{f}^{\text{'}}\left(s\right)|\le {C}_{2}{\left(1+|s|}^{p-2}\right)$.

It is shown that the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ generated by this problem possesses a global attractor $𝒜$ in ${L}^{1}\left({\Omega }\right)$ which is invariant, compact in ${L}^{p-1}\left({\Omega }\right)\cap {W}_{0}^{1,q}\left({\Omega }\right)$ with

$q

and attracting every bounded subset of ${L}^{1}\left({\Omega }\right)$ in the norm of ${L}^{r}\left({\Omega }\right)\cap {H}_{0}^{1}\left({\Omega }\right)$ with $r\in \left[1,+\infty \right)$. 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 $w\left(x,t\right)=u\left(x,t\right)-v\left(x\right)$ which satisfies

$\left\{\begin{array}{cc}{w}_{t}-{\Delta }w=f\left(v\right)-f\left(v+w\right)\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }×{ℝ}^{+},\hfill \\ w\left(x,0\right)={u}_{0}\left(x\right)-v\left(x\right)\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \\ w\left(x,t\right)=0\hfill & \text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }×{ℝ}^{+},\hfill \end{array}\right\$

where $v$ satisfies the elliptic equation $-{\Delta }v+f\left(v\right)=g\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }$ with homogeneous Dirichlet boundary condition and $v\in {W}_{0}^{1,q}\left({\Omega }\right)$ for $q.

##### MSC:
 35B41 Attractors (PDE) 35B65 Smoothness and regularity of solutions of PDE 35K20 Second order parabolic equations, initial boundary value problems 35K57 Reaction-diffusion equations 35K58 Semilinear parabolic equations