This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established f...

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation containing functional dependence on the unknown functions. The existence and some properties of solutions in (0,∞) will be proved. Introduction In this work we consider initial-boundary value problems for the system

where 00, Z>0, and uq(x) >0, Vx G [—1, Z]. Without loss of generality, we may assume that uq(x) is smooth and bounded above by 1 such that uo(±Z) = 1. Since uo(x) is positive, the local (in time) existence and uniqueness of a classical solution of the problem (1.1)—(1.3) are trivial (see [8]). Many results in quenching, such as single point quenching and profiles, are similar to those b...

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t → +∞. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t → +∞, with the sam...

Approximation theorems, analogous to known results for linear elliptic equations, are obtained for solutions of the heat equation. Via the Cole-Hopf transformation, this gives rise to approximation theorems for a nonlinear parabolic equation, Burgers’ equation.

Solvability of forward–backward stochastic di erential equations with nonsmooth coe cients is considered using the Four-Step Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution t...