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

We establish the global well-posedness of the $D(A)-$valued strong solution to a nonlinear heat equation with constraints on a $\mathit{\text{Poincar}}\acute{\mathit{\text{e}}}\mathit{\text{domain}}$ $\text{bO}\subset {\mathbb{R}}^d$ 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 $T_u\text{bM}$, where $\mathcal{M}:=\{u\in L^2(\text{bO}):\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\text{bO})}=1\}$ is a submanifold of $L^2(\text{bO})$. The nonlinearity exponent satisfies $2\le p<\mathrm{\infty }$ for $1\le d\le 4$ 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 $m-\mathit{\text{accretive}}$evolution equations, we prove the existence of a global strong solution. Using {resolvent-idea } and the $\mathit{\text{Yosida}}\mathit{\text{approximation}}$ method, we derive regularity results. In the asymptotic analysis, $\text{bO}$ 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 $\mathit{\text{Łojasiewicz-Simon}}\mathit{\text{gradient}}\mathit{\text{inequality}}$ on a Hilbert submanifold [F. Rupp, $\mathit{\text{J.}}\mathit{\text{Funct.}}\mathit{\text{Anal.}}$, 279(8), 2020], to demonstrate that the unique global strong solution converges in $W^{2,q}(\text{bO})\cap W_0^{1,q}(\text{bO})$ to a stationary state, where $2\le q<\frac{2d}{d+4-4\beta }$ and $1<\beta <\frac{3}{2}$. 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.