Counter fluxes of solutions of degenerate parabolic equations (Q2747628)

The author studies the following boundary value problem: NEWLINE\[NEWLINE \begin{gathered} Lu \equiv -u_t\operatorname {sgn} u + a(x,t,|u|,u_x)u_{xx} + b(x,t,|u|,u_x) = 0;\\ u(k,t) = 0,\quad (-1)^ku(x,kl) = u_k(x) > 0,\quad 0 < x < 1,\quad k = 1,2 \end{gathered} NEWLINE\]NEWLINE in the domain \(\Omega = (0,l)\times (0,1)\). This mathematical model arises in the context of investigation of the so-called counter fluxes of some physical characteristics, for example, temperature, concentration of a substance, velocity of a fluids and others. The equation \(Lu = 0\) is of forward-backward parabolic type. To solve the above problem the author uses the method of elliptic regularization and obtains a priori estimates which are uniform with respect to the parameter of the regularization. The latter enables us to realize a passage to the limit, and as a result, to prove the existence of generalized solution to the original problem.