Global well-posedness and Asymptotic analysis of a nonlinear heat equation with constraints of finite codimension

TL;DR

This paper establishes the global existence, uniqueness, and asymptotic behavior of solutions to a constrained nonlinear heat equation in any dimension, extending previous results to more general domains and nonlinearities.

Contribution

It proves well-posedness and asymptotic convergence for a nonlinear heat equation with constraints on bounded domains, generalizing prior work to higher dimensions and damping nonlinearities.

Findings

Solutions exist globally and are unique in the specified function spaces.

The constraint manifold remains invariant under the flow.

Solutions converge to the ground state as time approaches infinity.

Abstract

We prove the global existence and the uniqueness of the $L^p\cap H_0^1-$valued ($2\leq p < \infty$) strong solutions of a nonlinear heat equation with constraints over bounded domains in any dimension $d\geq 1$. Along with the \textit{Faedo-Galerkin} approximation method and the compactness arguments, we utilize the monotonicity and the hemicontinuity properties of the nonlinear operators to establish the well-posedness results. In particular, we show that a Hilbertian manifold $\mathbb{M}$, which is the unit sphere in $L^2$ space, describing the constraint is invariant. Finally, in the asymptotic analysis, we generalize the recent work of [P. Antonelli, et. al. \emph{Calc. Var. Partial Differential Equations}, 63(4), 2024] to any bounded smooth domain in $\mathbb{R}^d$, $d\geq1$, when the corresponding nonlinearity is a damping. In particular, we show that, for positive initial datum and any $2\le p < \infty$, the unique positive strong solution of the above mentioned nonlinear heat equation with constraints converges in $L^p\cap H_0^1$ to the unique positive ground state.