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 $M$?



Convergence of heat flow on non-compact manifolds?
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 $m\in\mathbb N$.


