# weak solution to non-homogeneous initial-boundary value problem

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.