Solvability of Dirichlet problem of second order linear parabolic equations in spaces \(L_p\)
Under the minimal consistent condition the author proves the solvability in Liouville, Besov and Nikolskii spaces of the following Dirichlet problem in the bounded region \(\Omega=\{(x,t)|\;x\in G, T\in(0,t]\}\): NEWLINE\[NEWLINE{\partial\over\partial t}u(x,t)=\sum\limits_{i,j=1}^{N} {\partial\over\partial x_{i}}\left[a_{ij}(x,t) {\partial\over\partial x_{j}}u(x,t)\right]+\sum\limits_{j=1}^{N} b_{j}(x,t){\partial\over\partial x_{j}}u(x,t)+c(x,t)u(x,t)+f(x,t),NEWLINE\]NEWLINE with the initial and boundary conditions \(u(x,0)=\phi(x)\), \(u(x,t)|_{\partial G}=\psi(x,t)\). In the Liouville case \(L_{p}\) this minimal consistent condition has the form \(\phi(x)-\widetilde\psi(x,t)\sim (t^{2}+w^{2}(x))^{\gamma}\), \(\gamma\in ({2p-3\over 2p},1]\), as \(t\to 0, x\to\partial G\), where the function \(w(x)\) satisfies some conditions on the boundary \(\partial G\) and \(\psi(x,t)\) is a trace of the function \(\widetilde\psi(x,t)\) on the surface \(\partial G\).
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