I am studying the theory of parabolic partial differential equation of type $u_t-\Delta u=f$. Now while defining the weak formulation we have used the Bochner spaces $L^2(0,T;H_0^1(\Omega))$. Can anyone explain why we consider these vector valued spaces to study parabolic equations. Can't we take functions of $(x,t)$ directly?
2026-03-26 12:41:19.1774528879
motivation behind using Bochner spaces in study of time dependent parabolic problems
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