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arXiv · 2210.04603

Existence and asymptotic behavior for $L^2$-norm preserving nonlinear heat equations

Paolo Antonelli·Piermarco Cannarsa·Boris Shakarov
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Abstract

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the $L^2$-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in $H^1$. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive.

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Paolo Antonelli, Piermarco Cannarsa, Boris Shakarov. 2022-10-10. Existence and asymptotic behavior for $L^2$-norm preserving nonlinear heat equations. https://doi.org/10.1007/s00526-024-02724-6

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