Is there a good reference for the following problem? Consider any smooth bounded domain $\Omega$ and solve the heat equation


\begin{align}
 \partial\_t u^\kappa &= \kappa \Delta u^\kappa,\\
 u^\kappa|\_{\partial\Omega}&=0,\\
 u^\kappa|\_{t=0}&=u\_0.
 \end{align}


What is the rate of convergence as $\kappa\to 0$ of $u^\kappa(t)$ to $u\_0$?



Rate convergence of the heat equation as diffusion tends to zero
**In a nutshell:** The rate of convergence depends on the smoothness of $u\_0$.


Here are the details.


**Setting:** 


* Throughout I assume that $\kappa > 0$ is a real number, i.e. it does not depend on the spatial variable.
* I'm going to consider the equation in the space $L^2(\Omega)$ (but similar observations also hold on $L^p(\Omega)$ for $p \in [1,\infty)$).
* Nomenclature: For functions $f: (0,\infty) \to L^2(\Omega)$ and $r: (0,\infty) \to (0,\infty)$ I will write $f \in \mathcal{O}(r)$ if there exists a real number $c \ge 0$ such that $\|f(s)\| \le c r(s)$ for all sufficiently small $s \in (0,\infty)$ (where $\|\cdot\|$ denotes the norm in $L^2(\Omega)$).
* Let $A$ denote the realisation of the Dirichet Laplace operator on $L^2(\Omega)$, i.e.
\begin{align\*}
 D(A) & = \{u \in H^1\_0(\Omega): \; \Delta u \in L^2(\Omega)\}, \\
 Au & = \Delta u;
\end{align\*}
here $\Delta u$ is to be understood in the sense of distributions. Note that $D(A)$ coincides with $H^2(\Omega) \cap H^1\_0(\Omega)$ if $\Omega$ has $C^2$-boundary (by elliptic regularity theory).
* The operator $\kappa A$ (with domain $D(\kappa A) = D(A)$) generates a $C\_0$-semigroup $(e^{t \kappa A})\_{t \ge 0}$; each operator $e^{t \kappa A}$ has norm at most $1$. For every $u\_0 \in L^2(\Omega)$ the (mild) solution of the heat equation in the question is given by the mapping
\begin{align\*}
 [0,\infty) \ni t \mapsto e^{t \kappa A}u\_0 \in L^2(\Omega);
\end{align\*}
this is even a classical solution if $u\_0 \in D(A)$.


To answer the question we can use the following idea: If we fix a time $t \in (0,\infty)$, then the behaviour of $e^{t\kappa A}$ as $\kappa \to 0$ is the same as the behaviour of $e^{sA}$ as $s \to 0$ (set $s := t\kappa$); thus, the question is essentially "how continuous" the mapping $s \mapsto e^{sA}u\_0$ is at $s = 0$.


Now, the answer consists of a negative and a positive result:


**Theorem 1** (negative result). There is no rate which works for all $u\_0 \in L^2(\Omega)$. More precisely: Fix $t \in (0,\infty)$. For every continuous function $r: (0,\infty) \to (0,\infty)$ which fulfils $r(s) \to 0$ as $s \to 0$ there exists a vector $u\_0 \in L^2(\Omega)$ such that the mapping $\kappa \mapsto e^{t\kappa A}u\_0 - u\_0$ is not contained in $\mathcal{O}(r)$.


*Proof.* Assume to the contrary that we can find a continuous function $r: (0,\infty) \to (0,\infty)$ which fulfils $r(s) \to 0$ as $s \to 0$ as well as $\big(\kappa \mapsto e^{t\kappa A}u\_0 - u\_0\big) \in \mathcal{O}(r)$ for each $u\_0 \in L^2(\Omega)$. Then it follows from the uniform boundedness principle that the operator family $\big((e^{t \kappa A} - \operatorname{id})/r(\kappa)\big)\_{\kappa \in (0,1]}$ is bounded with respect to the operator norm (here, $\operatorname{id}$ denotes the identity operator on $L^2(\Omega)$). Hence, $e^{t\kappa A}$ converges to $\operatorname{id}$ with respect to the operator norm as $\kappa \to 0$. Consequently, $e^{sA}$ converges to $\operatorname{id}$ with respect to operator norm as $s \to 0$. By standard $C\_0$-semigroup theory this can only be true if $A$ is a bounded operator, which is not fulfilled. This proves Theorem 1.


**Theorem 2** (positive result). Let $u\_0 \in D(A)$. Then
\begin{align\*}
 \|e^{t\kappa A} u\_0 - u\_0\| \le t\kappa \|\Delta u\_0\|
\end{align\*}
for all $t,\kappa \in (0,\infty)$. Hence, the convergence of $e^{t\kappa A}u\_0$ to $u\_0$ as $\kappa \to 0$ has a linear rate.


*Proof.* Fix $t,\kappa \in (0,\infty)$. Since $u\_0 \in D(A)$, we have
\begin{align\*}
 e^{t\kappa A}u\_0 - u\_0 = \int\_0^{\kappa t} e^{sA}A u\_0 \, ds
\end{align\*}
(see for instance [Engel/Nagel: One-Parameter Semigroups for Linear Evolution Equations (2000), Lemma II.1.3(iv)]). The assertion now readily follows by taking the norm on both sides of this equation (use that $Au\_0 = \Delta u\_0$ and that $\|e^{sA}\| \le 1$ for each $s$).


