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Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities. (English) Zbl 1077.35031
In this paper, the authors consider the following abstract semilinear parabolic equation in a Banach space $$X$$, $\frac{du}{dt}(t)+Au(t)=F(u(t)),\qquad u(0)=u_{0}\in X.\tag{1}$ where $$A:D(A)\subset X\rightarrow X$$ is a sectorial operator with $$\operatorname{Re}\sigma(A)>0,$$ $$F:X^{1}\rightarrow X^{\alpha},\alpha\geq0,$$ is a locally Lipschitz continuous map, and $$X^{\alpha},\alpha\geq0,$$ denote the fractional power spaces associated to $$A.$$ A mild solution to (1) is a function $$u(.,u_{0})\in C([0,\tau_{u_{0}}),X^{1})$$ which satisfies $u(t,u_{0})=e^{-At}u_{0}+ \int_{0}^{t}e^{-A(t-s)}F(u(s,u_{0}))\,ds.\tag{2}$ If $$F$$ does not take $$X^{1}$$ into $$X^{\alpha}$$ for any $$\alpha>0,$$ we say $$F$$ is critical; in this case, we define $$\varepsilon$$-regular solutions to (1) as functions in $$C([0,\tau),X^{1})\cap C((0,\tau),X^{1+\varepsilon})$$ which satisfy (2), and we say that (1) is locally well posed if for each $$u_{0}\in X^{1},$$ there is a unique $$\varepsilon$$-regular solution to (1) defined on a maximal interval of existence which depends continuously on the initial data $$u_{0}.$$ J. M. Arrieta and A. N. Carvalho [Trans. Am. Math. Soc. 352, No. 1, 285–310 (2000; Zbl 0940.35119)] studied the wellposedness of (1) when $$F$$ is critical. In this paper, the authors study the continuation properties and asymptotic behavior of $$\epsilon$$-regular solutions to (1) in the critical case, and apply their results to show the global solvability and the existence of a global attractor for a strongly damped wave equation in $$H_{0}^{1}(\Omega )\times L^{2}(\Omega).$$

MSC:
 35B41 Attractors 35K90 Abstract parabolic equations 35K55 Nonlinear parabolic equations 35L30 Initial value problems for higher-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations
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References:
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