### Electronic Research Archive

2020, Issue 3: 1357-1374. doi: 10.3934/era.2020072
Special Issues

# Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

• 35B40, 35B41, 37L30

• 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

### Related Papers:

• 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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