# Bounded solution for parabolic equation

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?

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 $$\|u_t\|_{L_\infty(\Omega_T)}\le C(T,a_0)(\|u_0\|_{L_\infty(\Omega)}+\|\Delta u_0\|_{L_\infty(\Omega)}).$$
• 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. – S. Maths Dec 26 '18 at 18:24
• 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/… – S. Maths Dec 26 '18 at 18:30
• 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. – Andrew Dec 26 '18 at 19:13