Convergence of heat flow on non-compact manifolds?

Question

Consider the heat equation $\partial_t u= \Delta u+\lambda_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda_1$ is the first eigenvalue and we start with some smooth initial data $u_0$ at $t=0.$ Then does the heat flow converge to the first eigenfunction on $M$?

Answer 1

The following answer only applies if $u_0\in L^2$, and it mostly relies on abstract operator theoretic arguments. I am sure more could be said by exploiting the geometric structure.

The Laplace-Beltrami operator on a complete manifold is essentially self-adjoint on $C_c^\infty$. Since $\lambda_1$ is the bottom of the spectrum of $-\Delta$, the operator $-(\Delta+\lambda_1)$ has a unique positive self-adjoint extension, which I denote by $A$. The heat flow is given by $e^{-tA}u_0$.

Let $E$ denote the spectral measure of $A$ and $d\mu(\lambda)=d\langle u_0,E(\lambda) u_0\rangle$. Note that $\mu$ is a finite measure on $\mathbb R_+$ with $\mu(\mathbb R_+)=\lVert u_0\rVert_2^2$. By the spectral theorem we have $$ \lVert A^k(e^{-tA}-1_{\{0\}}(A))u_0\rVert_2^2=\int_{[0,\infty)}\lambda^{2k}(e^{-t\lambda}-1_{\{0\}}(\lambda))^2\,d\mu(\lambda). $$ For $t\geq t_0$ (and $\lambda\geq 0$) we have $$ \lambda^{2k}(e^{-t\lambda}-1_{\{0\}}(\lambda))^2\leq \lambda^{2k}e^{-2t\lambda}\lesssim_k t_0^{-2k}. $$ Thus $\lVert A^k(e^{-tA}-1_{\{0\}}(A))u_0\rVert_2^2\to 0$ as $t\to\infty$ by the dominated convergence theorem.

The operator $1_{\{0\}}(A)$ is the projection onto $\ker(\Delta+\lambda_1)$. In particular, if $\lambda_1$ is an eigenvalue of $-\Delta$, then $1_{\{0\}}(A)u_0$ is an eigenfunction of $-\Delta$. In the following I will write $u_t$ for $e^{-tA}u_0$ and $u_\infty$ for $1_{\{0\}}(A)u_0$. By what we have seen before, $u_t\to u_\infty$ in $L^2$.

By elliptic regularity and Sobolev embedding, if $m\in\mathbb N$ and $2k\geq m+\dim(M)/2+1$, then $$ \lVert u_t-u_\infty\rVert_{C^m(\Omega)}\lesssim_{m,\Omega}\lVert u_t-u_\infty\rVert_{H^{2k}(\Omega)}\lesssim_{k,\Omega}\lVert A^k(u_t-u_\infty)\rVert_{L^2(\Omega)}+\lVert u_t-u_\infty\rVert_{L^2(\Omega)} $$ for every sufficiently nice bounded domain $\Omega$ in $M$.

Therefore, $u_t\to u_\infty$ in $C^m$ on compact subsets for every $m\in\mathbb N$.