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A simple unified approach to some convergence theorems of L. Simon. (English) Zbl 0895.35012
The author considers the semilinear evolution problems $u_{t} + Au = f(x,u), \qquad u(0,\cdot) = u_{0}(\cdot),\tag{1}$ $-u_{tt} + u_{t} + Au = f(x,u),\qquad u(0,\cdot) = u_{0}(\cdot),\qquad u_{t}(0,\cdot) = u_{1}(\cdot),\tag{2}$ and the associated stationary problem $Au = f(x,u), \tag{3}$ where $$A$$ is an operator of elliptic type, and $$f$$ satisfies an analyticity condition. He shows that, if a global solution $$u$$ of $$(1)$$ or $$(2)$$ is precompact in a suitable sense, then it converges, as $$t\to+\infty$$, to a solution of $$(3)$$. A result in the same direction was already proved in L. Simon [Ann. Math., II. Ser. 118, 525-571 (1983; Zbl 0549.35071)]. Here the author provides simplifications and extensions of that approach, that allow to consider also higher order operators $$A$$ and vector valued systems of gradient type.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35L70 Second-order nonlinear hyperbolic equations
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##### References:
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