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Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\in \Omega_T ,$$ $$u|_{\partial \Omega}=0,$$ $$u(0,\cdot)=u_0.$$ Assume that the initial condition $u_0 \in H^2(\Omega)\cap H^1_0(\Omega) \cap L^\infty(\Omega)$, can we prove that the solution $u$ is such that $u_t \in L^\infty(\Omega_T)$ ?

I found some old results which use much of regularity on $a$ and $u_0$ based on maximum principles. Are there any other ways to prove such results under weaker assumptions?

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For the heat equation $Lu=u_t-\Delta u=0$ to guarantee boundedness of $u_t$ as $t$ tends to zero one has to demand more regularity from the initial function, e.g. $u_0\in C^{1,1}(\bar\Omega)$ (the first order derivatives are uniformly Lipschitz in $\Omega$).

As for the low order term, differentiating wrt $t$ we have that $u_t$ satisfies the same equation. So if $u_0,\Delta u_0\in L_\infty(\Omega)$ then $u_t|_{t=0}\in L_\infty(\Omega)$ and $u_t$ is bounded.

Denote $a_0=\|a\|_{L_\infty(\Omega)}$ and $v(t)=\|u\|_{L_\infty(\Omega_t)}$. For the first BVP $u_t-\Delta u=f$, $u|_{t=0}=u_0$ with zero boundary condition it follows that $$ v(t)\le t a_0 v(t)+\|u_0\|_{L_\infty(\Omega)}. $$ From here for $T_0=1/(2a_0)$ it follows that $v(T_0)\le 2\|u_0\|_{L_\infty(\Omega)}$. For arbitrary $T$ the estimate $v(T)\le C\|u_0\|_{L_\infty(\Omega)}$ follows from step by step argument, where $C$ depends upon $T$ as well as on $a_0$. For the derivative it gives the estimate $$ \|u_t\|_{L_\infty(\Omega_T)}\le C(T,a_0)(\|u_0\|_{L_\infty(\Omega)}+\|\Delta u_0\|_{L_\infty(\Omega)}). $$

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  • $\begingroup$ Thank you @Andrew, I'm looking for further assumptions which imply the boundedness result. I think it is sufficient to see the boundedness of the solution and deduce the result for the time derivative. $\endgroup$
    – S. Maths
    Commented Dec 26, 2018 at 14:32
  • $\begingroup$ Okey, in my case $Lu=u_t- \Delta u=f=au$. What is the assumption on $f$ to obtain the boundedness for $u$. I need some references on such results. Thanks. $\endgroup$
    – S. Maths
    Commented Dec 26, 2018 at 18:24
  • $\begingroup$ I found that the assumption is $f\in L^\infty(\Omega_T)$ and this is the result we look for. See this link math.stackexchange.com/questions/776017/… $\endgroup$
    – S. Maths
    Commented Dec 26, 2018 at 18:30
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    $\begingroup$ For the heat equation in a smooth bounded domain there exists the Green function $G$ of the first BVP. The solution with zero initial condition can be written as $$ u(x,t)=\int_0^t \int_\Omega G(x,y,t-\tau)f(y,\tau)\,dyd\tau. $$ Also one can get the estimate $0<G(x,y,t)<Z(x-y,t)$, where $Z$ is the fundamental solution of the heat equation. From there the estimate $$ |u(x,t)|\le t \|f\|_{L_\infty(\Omega_t)} $$ follows immediately. $\endgroup$
    – Andrew
    Commented Dec 26, 2018 at 19:13
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    $\begingroup$ I've edited the answer. $\endgroup$
    – Andrew
    Commented Dec 27, 2018 at 18:27

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