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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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[2] Arrieta, J.M.; Carvalho, A.N., Abstract parabolic problems with critical nonlinearities and applications to navier – stokes and heat equations, Trans. amer. math. soc., 352, 285-310, (2000) · Zbl 0940.35119
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