Similarity solutions for a quasilinear parabolic equation

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption

Quenching Profile for a Quasilinear Parabolic Equation

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...

Rational Approximation for a Quasilinear Parabolic Equation

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.

Positive solutions for a quasilinear Schrödinger equation

We consider the quasilinear problem −εpdiv(|∇u|p−2∇u) + V (z)up−1 = f(u) + up−1, u ∈W (R ), where ε > 0 is a small parameter, 1 < p < N , p∗ = Np/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory...

Self-similar solutions in a sector for a quasilinear parabolic equation

We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally “self-similar” in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is selfsimilar at discrete times. We also study the existence and uniqueness of a shrinking solution which is sel...