New user here, please pardon my mistakes. During my research I was faced with the following type of variational equation
$$ \int_0^T \int_{\Omega} \nabla u \cdot \nabla v -\partial_t u \partial_t v + \partial_t u c(t,x) \cdot \nabla v + \partial_t v d(t,x) \cdot \nabla u dx dt $$
where $\Omega \subset \mathbb{R}^n$ is a bounded and Lipschitz domain, the functions $u,v \in H^1_0(\Omega)$ and the functions $c,d \in C([0,t]\times \Omega;\Omega)$.
What I need are Existence and Uniqueness theorems for these kinds of equations, so I looked for the classical or strong formulation of such problems, but could only find papers for the 1-dimensional case under the name of first mixed problem.
Morou, A.F., Gbenouga, N., Tcharie, K., "On the classic solution of the first mixed problem for some general hyperbolic equations one-dimensional of the second order with souce term", Global Journal of Pure and Applied Mathematics, 2016, Vol. 12, Nº 6, pp. 5175–5184
Baranovskaya, S.N., “On the classical solution of the mixed problem for one di- mensional hyperbolic equation”, Differential equation (CEI), 1991, T.27, N◦ 6, pp. 1071–1073.
Baranovskaya, S.N. Tcharie Kokou, “On the classical solution of the mixed problem for a general one dimensional hyperbolic equation”, Differential equation (CEI), 1999, T.35, N◦ 8, pp.
I am looking for something on the $n$-dimensional case, since in these papers they seem to use techniques that rely heavily in the dimensionality of the problem. My aim is to apply techniques from Sobolev space theory or maybe even Semigroup theory.
Thanks for any answers.
Edit: typo