Abstract
The wellposedness and energy estimate for wave equations in domains with a space-like boundary
Highlights
QkT is named time like, if the inequality |nt(p)| < |nx(p)| holds for every point p ∈ Σ
It is easy to see that wave equations are defined in the domain QkT with a space-like boundary ΓR
For wave equations in space-like domains, [4] was the one and only one paper we have known providing a condition that solutions and all their first-order derivatives vanish on Σ to make systems well-posed
Summary
It is easy to see that wave equations are defined in the domain QkT with a space-like boundary ΓR. There are many literatures on wave equations in non-cylindrical domains with time-like boundary For wave equations in space-like domains, [4] was the one and only one paper we have known providing a condition that solutions and all their first-order derivatives vanish on Σ to make systems well-posed. (2) From (1.2), we claim that such a problem with ordinary Dirichlet boundary conditions may be ill-posed (Since f3 is given freely in an appropriate function space, we can choose different f3, but keep f1 and f2 the same, and the system has multiple solutions). For the case of time-like domains, we mention [3, 11], which provided some first-order polynomial decay results using the multiplier method. Since the above inequality holds for any τ1 ∈ (0, T1], integrating it on (0, T1), we obtain
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