Well-posedness and the Łojasiewicz-Simon inequality in the asymptotic analysis of a nonlinear heat equation with constraints of finite codimension

Abstract: We establish the global well-posedness of the $D(A)-$valued strong solution to a nonlinear heat equation with constraints on a \textit{Poincaré domain} $\bO\subset \R^d$ whose boundary is of class $C^2$. Consider the following nonlinear heat equation \begin{align*} \frac{\partial u}{\partial t} - Δu + |u|^{p-2}u = 0, \end{align*} projected onto the tangent space $T_u\bM$, where $\mathcal{M}:=\left{u\in L^2(\bO):|u|_{L^2(\bO)}=1\right}$ is a submanifold of $L^2(\bO)$. The nonlinearity exponent satisfies $2\le p < \infty$ for $1\leq d\leq 4$ and $2 \le p \le \frac{2d-4}{d-4}$ for $d \ge 5$. The solution is constrained to lie within $\mathcal{M}$ which encodes the norm-preserving constraint. By modifying the nonlinearity and exploiting the abstract theory for \textit{$m-$accretive }evolution equations, we prove the existence of a global strong solution. Using {resolvent-idea } and the \textit{Yosida approximation} method, we derive regularity results. In the asymptotic analysis, $\bO$ is restricted to bounded domains with even $p$ and $1\le d \le 3$. For any initial data in $D(A) \cap \mathcal{M}$, we apply the \textit{Łojasiewicz-Simon gradient inequality} on a Hilbert submanifold [F. Rupp, \textit{J. Funct. Anal.}, 279(8), 2020], to demonstrate that the unique global strong solution converges in $W^{2,q}(\bO) \cap W^{1,q}_0(\bO)$ to a stationary state, where $2 \le q < \frac{2d}{d + 4 - 4β}$ and $1 < β< \frac{3}{2}$. This work proposes an alternative method for establishing the global existence and analyzing long-term behavior of the unique strong solution to an $L^2-$norm preserving nonlinear heat equation.