Let $\Omega\subset \mathbb{R}^n$ be a bounded Lipschitz domain, and consider a general linear parabolic equation of the divergence form: $$ u_t-\partial_i(a^{ij}\partial_ju)+b^i\partial_i u+cu=f, \quad (t,x)\in (0,T)\times\Omega ,\tag{1} $$ with $u(0,x)=g(x)$ for $x\in \Omega$ and $u(t,x)=h(t,x)$ for $(t,x)\in (0,T)\times \partial\Omega$. Assume $a^{ij},b^i,c\in L^\infty(\Omega)$, and $(a^{ij})_{ij}$ is uniformly elliptic.

It has been shown in Theorem 6.3 that if $h\equiv 0$, then for any given $f\in L^2(0,T; H^{-1}(\Omega))$ and $g\in L^2(\Omega)$, (1) has a weak solution with $u\in C(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$ and $u'\in L^2(0,T; H^{-1}(\Omega))$.

I was wondering, whether there is a corresponding theorem for nonhomogeneous $h$, say $h\in L^2(0,T;H^{1/2}(\partial\Omega))$?

I guess it is related to the trace theorem of time-dependent Sobolev space. But I cannot find any result on it.


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