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On the Łojasiewicz–Simon gradient inequality. (English) Zbl 1036.26015
The author establishes in this paper a version of the Łojasiewicz-Simon inequality for functions which are not necessarily analytic. The abstract result is then applied to energy functionals of the form \(E(v)=\frac {1}{2}a(v,v)+\int_\Omega F(x,v)\,dx\), defined on a Hilbert space which is compactly embedded into \(L^2(\Omega)\). In the last part of the paper these results are applied to nonlinear evolution equations of the form \(u_t-\Delta u+| u| ^{p-1}u+\lambda u=0\) in \({\mathbb R}\times\Omega\). It is shown that in this example the underlying energy functional satisfies the Łojasiewicz-Simon inequality near the origin with Łojasiewicz exponent \(\theta =(p+1)^{-1}\). The proofs are based on refined estimates and a careful analysis of several classes of differential operators.

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
34D05 Asymptotic properties of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
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