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Uniform quasi-differentiability of semigroup to nonlinear reaction-diffusion equations with supercritical exponent. (English) Zbl 1389.37046

Summary: A new approach is established to show that the semigroup \({\left\{ {S\left( t \right)} \right\}_{t \geq 0}}\) generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in \({L^q}\left( \Omega \right)\left( {2 \leq q < \infty} \right)\) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35K57 Reaction-diffusion equations
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