Solvability of a parabolic boundary value problem in the presence of power singularities in the right sides. (Russian) Zbl 0657.35067

Let G be a smooth bounded domain in \(R^ n\), \(\Omega =\bar G\times [0,T)\), \(b>0\) an integer, \(S=\partial G\times [0,T)\), \({\mathcal H}^ s(\Omega)={\mathcal H}^ s_ x{}_ t^{s/2b}(\Omega)\) and \({\mathcal H}^ s(S)={\mathcal H}_ x^ s{}_ t^{s/2b}(S)\) (anisotropic Sobolev- Slobodetskii’s spaces), P a subset of \(Z\times Z\), \(Z_ a=\{z\in Z:\) \(z>0\), \(z/a=\) (mod 1)\(\}\). For the definition of spaces \(\tilde H^ s(G)\) and \(\tilde {\mathcal H}_{(\alpha,\tau,P)}(G)\) \((s\not\in Z_ 1\cup Z_{2b})\) see the present author [Math. USSR, Sb. 56, 447-471 (1987); translation from Mat. Sb., Nov. Ser. 128(176), No.4, 451-473 (1985; Zbl 0609.35045)]. Let \({\mathcal L}(x,t,D,D_ t)\) be a parabolic operator in \({\bar \Omega}\) of order \(2m=2br\), \(b_ j(x',t,D,D_ t)\) \((j=1,...,m)\) operators of weighted order \(m_ j<2m\) connected with L on S by the Lopatinskii’s (complementarity) condition. Let us consider the following problem: Find a solution u in \(\tilde {\mathcal H}^ s_{(r,2m,P)}(\Omega)\) to the boundary value problem \[ {\mathcal L}(x,t,D,D_ t)u(x,t)=f(x,t)\in {\mathcal H}^{s-2m}(\Omega), \]
\[ {\mathcal B}_ j(x',t,D,D_ t)u(x,t)_{| S}=\phi_ j(x',t)\in {\mathcal H}^{s-m_ j-}(S)\quad (j=1,...,m), \]
\[ D_ t^{\lambda -1}u(x,t)_{| t=0}=\psi_{\lambda}(x)\in \tilde H_{(P_{\lambda})}^{s-2b\lambda +b}(G)\quad (\lambda =1,...,r), \] where \(P_{\lambda}=\{k: (k,\lambda)\in P\}.\)
If \(f(x,t)=h(x,t)\) \((| y'|^{2b}+| t-t_ 0|)^{-\mu}\), where \(h\in L_ 2(\Omega)\), 2b\(\mu\geq q/2+b\), \(F_{\mu}(x,t)=reg f(x,t)\) is a regularization of f(x,t) with \(s=2b\mu -q/2-b+\epsilon\) \((\epsilon >0)\), \(\phi_ j\in {\mathcal H}^{2m-2b\mu -m_ j- +\epsilon}(S),\) \(\psi_{\lambda}(x)\in \tilde H^{2m-2b\lambda -2b\mu +b+\epsilon}(G)\) and the compatibility condition is satisfied for \(t=0\), then the problem \[ {\mathcal L}u=F_{\mu}(x,t),\quad {\mathcal B}_ ju_{| S}=\phi_ j(x',t)\quad (j=1,...,m),\quad D_ t^{\lambda - 1}u(x,0)=\psi_{\lambda}(x)\quad (\lambda =1,...,r) \] is solvable and the solution lies in \(\tilde {\mathcal H}^{2m-2b\mu - \epsilon}_{(r,2m,P)}(\Omega)\).
Reviewer: J.Danesova


35K35 Initial-boundary value problems for higher-order parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems


Zbl 0609.35045
Full Text: EuDML