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Consider the following problem: $$ \begin{cases} \mu(\mathbf{x}) \partial_t y(\mathbf{x},t) - \Delta y(\mathbf{x},t) = f(\mathbf{x},t) \qquad \text{in} \ \Omega \times (0,T) \\ y=0 \qquad \text{on} \ \partial \Omega \times (0,T) \\ y(\cdot,0)=y_0 \qquad \text{in} \ \Omega \end{cases}$$ where $\mu \in L^{\infty}(\Omega)$ is such that there exists a strictly positive real number $\mu_0$ satisfying $\mu(\mathbf{x}) \ge \mu_0 $ almost everywhere.

I want to study the problem using abstract variational Hilbert spaces techniques for evolution equations, after some work I end up with the following weak problem:

$V:= H^1_0(\Omega) $, $H := L^2(\Omega) $ and we have the well-known Hilbert (Gelfand) triple $ V \hookrightarrow H \hookrightarrow V' $, with dense injections. $H$ is endowed with the scalar product $$ (u,v)_H := \int_{\Omega} \mu u v \, dx, $$ which is equivalent to the standard one under the given hypothesis. We now assume $f \in L^2(0,T ; V') $ and rewrite our problem as (remember that derivatives are generally understood in a distributional sense)

Find $y \in W_{\mu}(0,T):= \{ u \in L^2(0,T;V) : u' \in L^2(0,T; V') \} $:

$$ \frac{d}{dt} (y(t),v)_{H} + a(y(t),v) = \langle f(t),v \rangle_{V' \times V} $$ for all $ v \in V$ and almost every $t \in (0,T)$. Moreover $$ y(0)=y_0 \in H $$ Having defined $a(\cdot, \cdot) $ as the usual gradient-gradient bilinear form. Existence and uniqueness for this problems are guaranteed by known theorems (see references), provided the bilinear form satisfies some requests.

The problem I have is to clarify the role of $\mu$ a posteriori/get a meaningful strong interpretation: the definition of a weighted scalar product in $H$ has been made to incorporate $\mu$ in the weak formulation. As existence is guaranteed, we have proven, in some sense, that $$ (\mu y)_t = \mu y_t \in L^2(0,T;V'), $$ does this imply that $y_t \in L^2(0,T;V')$, as we would have if we were considering a standard heat equation without the coefficient $\mu$? A tempting trick would be to substitute $z := \mu y$ since the beginning, but then we would need some regularity of $\mu^{-1} v$ in order to set up the weak formulation! This confuses me a bit, especially because the standard (and formal) strong form I wrote above is equivalent to the one we get performing a pointwise multiplication by $\mu^{-1}$.

Thanks in advance for every suggestion.

Refereces:

  • Dautray, Robert, and Jacques-Louis Lions. Mathematical analysis and numerical methods for science and technology: volume 5, Evolution problems. Springer Science & Business Media, 2012. (page 512)

  • Zeidler, Eberhard. Nonlinear functional analysis and its applications: II/A: Linear and monotone operators. Springer Science & Business Media, 2013. (page 422-424)

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