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Global existence and maximal regularity of solutions of gradient systems. (English) Zbl 1209.47020

The author applies the Galerkin method in order to prove a maximal regularity result for the following abstract gradient system
\[ \begin{cases} u'(t)+\nabla_g E(u(t))=f(t)\quad \text{for a.a.}\;t\in[0,T],\\ u(0)=u_0. \end{cases} \]
Here, \(E: V\to\mathbb R\) is a continuously differentiable function over a real Banach space \(V\) densely and compactly embedded into a real Hilbert space \(H,\) \(\nabla_g E\) stands for the gradient of \(E\) with respect to some metric \(g,\) \(f\in L^2(0,T;H)\) and \(u_0\in V.\)
The abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.

MSC:

47E05 General theory of ordinary differential operators
35Q99 Partial differential equations of mathematical physics and other areas of application
35B65 Smoothness and regularity of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
47A60 Functional calculus for linear operators
47J35 Nonlinear evolution equations
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References:

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