In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $ p>2 $ order nonlinearity and in any space dimension $ N \geqslant 1 $. It is proved that the weak solutions can be $ (L^2, L^\gamma\cap H_0^1) $-continuous in initial data for arbitrarily large $ \gamma \geqslant 2 $ (independent of the physical parameters of the system), i.e., can converge in the norm of any $ L^\gamma\cap H_0^1 $ as the corresponding initial values converge in $ L^2 $. In fact, the system is shown to be $ (L^2, L^\gamma\cap H_0^1) $-smoothing in a H$ \ddot{\rm o} $lder way. Applying this to the global attractor we find that, with external forcing only in $ L^2 $, the attractor $ \mathscr{A} $ attracts bounded subsets of $ L^2 $ in the norm of any $ L^\gamma\cap H_0^1 $, and that every translation set $ \mathscr{A}-z_0 $ of $ \mathscr{A} $ for any $ z_0\in \mathscr{A} $ is a finite dimensional compact subset of $ L^\gamma\cap H_0^1 $. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order $ p>2 $ of the nonlinearity and the space dimension $ N \geqslant 1 $.
Citation: Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension[J]. Electronic Research Archive, 2020, 28(3): 1357-1374. doi: 10.3934/era.2020072
In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $ p>2 $ order nonlinearity and in any space dimension $ N \geqslant 1 $. It is proved that the weak solutions can be $ (L^2, L^\gamma\cap H_0^1) $-continuous in initial data for arbitrarily large $ \gamma \geqslant 2 $ (independent of the physical parameters of the system), i.e., can converge in the norm of any $ L^\gamma\cap H_0^1 $ as the corresponding initial values converge in $ L^2 $. In fact, the system is shown to be $ (L^2, L^\gamma\cap H_0^1) $-smoothing in a H$ \ddot{\rm o} $lder way. Applying this to the global attractor we find that, with external forcing only in $ L^2 $, the attractor $ \mathscr{A} $ attracts bounded subsets of $ L^2 $ in the norm of any $ L^\gamma\cap H_0^1 $, and that every translation set $ \mathscr{A}-z_0 $ of $ \mathscr{A} $ for any $ z_0\in \mathscr{A} $ is a finite dimensional compact subset of $ L^\gamma\cap H_0^1 $. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order $ p>2 $ of the nonlinearity and the space dimension $ N \geqslant 1 $.
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Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension[J]. Electronic Research Archive, 2020, 28(3): 1357-1374. doi: 10.3934/era.2020072
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