# Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $$\zeta, u_0\in L^2(\Omega)$$, with $$\zeta \geq 0$$ and $$\Omega\subset \Bbb R^d$$ open and bounded.
$$$$\label{Star-3.7} \begin{cases} \partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\; \Omega\times (0, T),\\ u = 0 &\mbox{ in }\; \partial\Omega\times (0, T), \\ u(\cdot,0) = u_{0}, &\mbox{ in }\; \Omega, \end{cases}$$$$ We say that $$u: \Omega\times (0, T)\to \Bbb R$$ is a weak solution if: $$u \in L^2(0,T; H_0^1(\Omega) \cap L^2(\Omega, \zeta dx))$$ and we have $$\int_\Omega\partial_t uv +\int_\Omega\nabla u\nabla v+ \int_\Omega\zeta uv \qquad\text{ for all \quad v \in L^2(0,T; H_0^1(\Omega) \cap L^2(\Omega, \zeta dx))}.$$

Question: Is it possible to show the existence and uniqueness of a weak solution?

If Yes which method could be suitable here or what are the right references for such equations?

Here is a functional analytic approach (Kato's book on perturbation theory is a good reference):

Let $$a\colon D(a)\times D(a)\to \mathbb{R},\,(u,v)=\int \nabla u\cdot \nabla v+\int \zeta uv.$$ with $$D(a)=H^1_0(\Omega)\cap L^2(\zeta\,dx)$$. It is not hard to see that $$a$$ is closed, that is, $$D(a)$$ endowed with the inner product $$\langle\cdot,\cdot\rangle_2+a$$ is complete. Therefore there exists a positive self-adjoint operator $$A$$ on $$L^2(\Omega)$$ with $$D(A)\subset D(a)$$ such that $$a(u,v)=\langle Au,v\rangle$$ for $$u\in D(A)$$, $$v\in D(a)$$.

Let $$u(t,\cdot)=e^{-tA}u_0$$. By functional calculus, $$u\in C^\infty([0,\infty);H^1_0(\Omega)\cap L^2(\zeta\,dx))$$ and $$\partial_t u=-Au.$$ In particular, $$u$$ is a weak solution of your PDE by the definition of $$A$$.

To prove uniqueness, you can apply the usual energy method, i.e., show that $$a(u,u)$$ is decreasing along solutions.

Since you use it as measure, I guess $$\zeta$$ is non-negative ? Also your fortmulation needs to be integrated in time or you should add a regularity assumption for the time derivative.

I would first solve the equation in $$L^2(0,T;H^1_0(\Omega))$$ replacing $$\zeta$$ by $$\zeta_R:=\mathbf{1}_{|\zeta|\leq R} \zeta$$. In that case this just follows from standard parabolic theory (see Evans for instance).

Your then get a family of solution $$(u_R)_R$$ which solves a weak formulation in $$L^2(0,T;H^1_0(\Omega))$$. In particular, choosing $$u_R$$ as a test function in its own formulation you get after integration by parts

\begin{align*} \|u_R\|_{L^\infty(0,T;L^2(\Omega)}^2 + \|\nabla u_R\|_{L^2(0,T;L^2(\Omega)}^2 + \|u_R\|_{L^2(0,T;L^2(\Omega,\zeta_R \mathrm{d}x))}^2 \leq \|u_0\|_{L^2(\Omega)}^2. \end{align*}

Up to a subsequence $$(u_R)_R$$ is $$(\star$$-)weakly converging towards some $$u$$ in $$L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$$. Since $$(\xi_R)_R$$ converges in $$L^2(0,T;L^2(\Omega))$$, you can pass to the limit the equation, at least in the distribution sense.

To recover the weighted estimate, you can first get strong convergence for $$(u_R)_R$$ in $$L^2(0,T;L^2(\Omega))$$ by the Aubin-Lions lemma, because since $$(u_R\xi_R)_R$$ is bounded in $$L^1(0,T;L^1(\Omega))$$. Then, using \begin{align*} \int_0^T \int_\Omega \zeta_R |u_R|^2 \leq \int_\Omega |u_0|^2 \end{align*} you recover the weighted integrability by Fatou's lemma. The extended formulation follows by density.

• What is the aim behind the truncation argument? Do you intend to get $\zeta_R$ bounded? because the one you use is not bounded. Dec 19 '20 at 15:30
• Sorry, I corrected the typo. Dec 19 '20 at 16:28