# Regularity for a Nonhomogeneous Heat Equation

Let $$\Omega$$ be an open bounded subset of $$\mathbb{R}^n$$ with smooth boundary, and let $$T>0$$. Consider the non-homogeneous heat equation with Dirichlet boundary condition

\begin{aligned} u_t - \Delta u &= f & &\text{in }\Omega\times(0,T), \\ u &= 0 & &\text{on } \partial\Omega\times(0,T), \\ u(x,0) &= u_0(x) & &\text{for all } x \in \Omega.\end{aligned}

Suppose that $$f \in L^\infty(\Omega\times(0,T))$$ and $$u_0 \in L^\infty(\Omega)$$. Is it true that there exists a solution $$u$$ to the equation above such that $$u \in C^{2,1}(\bar{\Omega}\times(0,T)) \cap L^\infty(\Omega\times(0,T))$$ and $$\lim_{t \rightarrow 0} u(x,t) = u_0(x)$$ for a.e. $$x\in \Omega$$.

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First solution attempt. I tried to follow Section 2.3.1 in Evans book on pdes. But we only look at $$\Omega \subset \mathbb{R}^n$$, therefore we will use the Green function of the heat equation with Dirichlet boundary condition $$G$$ instead of the fundamental solution $$\Phi$$ that is used in the book. As in the book we split the problem into a homogeneous part with $$u_0$$ as initial data and a nonhomogeneous part with $$0$$ as initial data. Theorem 1 on page 47 in Evans should give us a solution for the homogeneous part. The proof for (i) and (ii) should still work for initial data in $$L^\infty$$, hence

$$u(x,t) = \int_\Omega G(x,y,t) u_0(y) \, \mathrm{d}y$$

is a smooth solution for the homogeneous part. Of course with noncontinuous initial data we can't expect (iii) to be true.

For the nonhomogeneous part we define

$$u(x,t) = \int_0^t\int_\Omega G(x,y,t-s) f(y,s) \, \mathrm{d}y \mathrm{d}s.$$

The problem is that in this case the regularity of $$u$$ doesn't follow straightforward from the regularity of $$G$$, because $$G$$ has a singularity at $$t=0$$, thus we can't differentiate under the integral. The proof of Theorem 2 on page 50 in Evans book assumes that $$f \in C^{2,1}(\Omega\times(0,T))$$ and that $$f$$ has compact support. The proof of Theorem 2 as presented by Evans doesn't work with $$f \in L^\infty$$. However, in Evans book it says that $$f \in C^{2,1}(\Omega\times(0,T))$$ with compact support is assumed for simplicity. The question is now if it is still possible to prove Theorem 2 with the assumption that $$f \in L^\infty$$ or does this approach just not work.

After writing down my first solution attempt I realized that it probably is not possible to find a regular $$u \in C^{2,1}$$ with just $$f\in L^\infty$$. The reason for this is that $$f$$ represents a heat source independent of $$u$$, thus we can't expect $$u$$ to be in $$C^{2,1}$$, if the outside heat source $$f$$ is non continuous.
Additionally, someone pointed out to me that if our solution $$u$$ is actually in $$C^{2,1}$$ then obviously $$u_t - \Delta u \in C^0$$. Hence, we need $$f$$ to be at least in $$C^0$$ if we want a solution $$u \in C^{2,1}$$. So we might be able to prove Evans Theorem 2 with $$f \in C^0$$ instead of $$f\in C^{2,1}$$. Nonetheless, we can say that the answer to my original question is no, because $$u \in C^{2,1}$$ implies $$f \in C^0$$ and hence $$f \in L^\infty$$ is not enough for the existence of a regular solution.