On some properties of solutions to parabolic equations with variable time direction (Q2713900)

Assume that \(G\) is a~bounded domain in \(\mathbb R^n\) with smooth boundary \(\Gamma\), \(Q=G\times(0,T)\), \(T\leq+\infty\), and \(L\) is the elliptic operator NEWLINE\[NEWLINELu=-\sum_{i,j=1}^n {\partial\over\partial x_i} \left(a_{ij}(t,x){\partial u\over\partial x_i}\right) +\sum^n_{i=1}b_i(t,x)u_{x_i}+c(t,x)u,\qquad a_{ij}=a_{ji}.NEWLINE\]NEWLINE Let the function \(g(t,x)\) be such that \(g,g_t\in L_\infty(0,T;L_p(G))\), \(p\geq n\) for \(n>2\), \(p=2\) for \(n=1\), \(p>2\) for \(n=2\). Assume further that there exist subsets \(G^+(0)\), \(G^+(T)\), \(G^-(0)\), and \(G^-(T)\) of \(G\) satisfying \(\mu((\overline{G^\pm(0)}\setminus G^\pm(0))=0\), \(\mu((\overline{G^\pm(T)}\setminus G^\pm(T))=0\) (with \(\mu\) the Lebegue measure), \(g(0,x)>0\) almost everywhere in \(G^+(0)\), \(g(T,x)>0\) almost everywhere in \(G^+(T)\), \(g(0,x)<0\) almost everywhere on \(G^-(0)\), \(g(T,x)<0\) almost everywhere on \(G^-(T)\), \(g(0,x)=0\) almost everywhere on \(G\setminus(\overline{G^+(0)\cup}\overline{G^-(0)})\), and \(g(T,x)=0\) almost everywhere on \(G\setminus(\overline{G^+(T)\cup}\overline{G^-(T)})\).NEWLINENEWLINENEWLINEDefine the operator \(B_1u\): NEWLINE\[NEWLINE B_1u=u(t,x)\Big|_{\Gamma_1}\text{ or } B_1u={\partial\over\partial N}u(t,x)+\sigma(t,x)u, NEWLINE\]NEWLINE where \(\Gamma_1=\Gamma\times(0,T)\), \({\partial\over\partial N}u=\sum^n_{i,j=1}a_{ij}u_{x_i}x_j\), \(n_j\) are components of the exterior unit normal to~\(\Gamma\).NEWLINENEWLINENEWLINEThe author considers the boundary value problem: Find a~solution in \(Q\) to the equation NEWLINE\[NEWLINE g(t,x)u_t+Lu=f(t,x) NEWLINE\]NEWLINE satisfying NEWLINE\[NEWLINEB_1u=0, \qquad u(0,x)=u_0(x)\quad\text{for } x\in G^+(0), \qquad u(T,x)=u_T(x)\quad\text{for } x\in G^-(T) NEWLINE\]NEWLINE in the case of \(T<+\infty\), and NEWLINE\[NEWLINE \lim_{t\to\infty}u(t,x)=0 NEWLINE\]NEWLINE in the case of \(T=+\infty\).NEWLINENEWLINENEWLINELet \(X_1\) be the space \(\overset\circ W^1_2(G)\) or \(W^1_2(G)\) in accordance with the kind of the operator \(B_1\), \(\varphi_i(t)=t^{2i}(T-t)^{2i}\) for \(t<+\infty\), \(\varphi_i(t)=\min(t^{2i},1)\) for \(T=\infty\), \(i=0,1,\dots\), \(B(t)\) be the operator of multiplication by \(g(t,x)\), \(X'_1\) be a~nonnegative space constructed on a~pair of the spaces \(X_1\) and \(L_2(G)\), and \(W^m\) be the spaces: NEWLINE\[NEWLINE\begin{multlined} W^m=\Biggl\{u\in L_2(0,T;X_1): \sqrt{\varphi_i}{\partial^i u\over\partial t^i}\in L_2(0,T;X_1), \quad i\leq m,\\ \sqrt{\varphi_m}{\partial^m u\over\partial t^m} \biggl(B(t){\partial^m u\over\partial t^m}\biggr) \in L_2(0,T;X'_1\Biggr\}, \quad m=0,1,\dots. \end{multlined}NEWLINE\]NEWLINE The author studies the properties of \(W^m\), gives sufficient conditions for a~solution of the boundary value problem to belong to the space \(W^0\), and proves some results about increasing internal smoothness of solutions.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].