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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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