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A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation
Authors:Dzhuraev  T. D.  Takhirov  J. O.
Affiliation:(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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