Nonlocal boundary-value problem for a nonlinear parabolic equation (Q2742901)

The nonlocal boundary-value problem for the nonlinear parabolic equation NEWLINE\[NEWLINEu_t=a(t,x,u,u_x,u_{xx})\tag{*}NEWLINE\]NEWLINE in the domain \(Q=\{(t,x)\mid 0\leq t\leq T\), \(|x|\leq l\}\subset\mathbb R^2\) is investigated. Nonlocal boundary conditions are taken in the form \(u(t,0)=0\), \(u(t,-l)=m_1u(t,-l_0)\), \(u(t,l)= m_2u(t,l_0)\), where \(l_0\in(0,l-2\delta]\), \(|m_1|\leq 1\), \(|m_2|\leq 1\). Under some conditions on the function \(a(t,x,u,p,r)\), to establish the global existence of the problem's solution a priori estimates are obtained. The uniqueness and existence theorems are stated.