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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.

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI
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