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A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

Authors:

Dzhuraev T D Takhirov J O

Institution:

(1) Institute of Mathematics, Academy of Sciences of Uzbekistan, 29, F. Hodjaev St., Tashkent, 700143, Uzbekistan

Abstract:

The solvability of the nonlocal boundary value problem

$\begin{gathered} u_t = a(t,x,u,u_x )u_{xx} + b(t,x,u,u_x ),{\text{ }}0 \leqslant t \leqslant T,{\text{ }}\left| x \right| \leqslant l, \hfill \\ {\text{ }}u(0,x) = 0,{\text{ }}u(t, - l) = u(t,l),{\text{ }}u_x (t, - l) = u_x (t,l) \hfill \\ \end{gathered}$

in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.

Keywords:

Quasilinear parabolic equations nonlocal boundary conditions a priori estimates existence of a solution

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