Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian

Abstract

The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p-Laplace operator p is investigated for p > 2. This problem is reduced to the comparison of the trivial solution (≡ 0, by hypothesis) with a nontrivial nonnegative solution u(x, t). The problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle. In this article, realistic counterexamples to the uniqueness of a nonnegative solution, the weak comparison principle, and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a Lipschitz nor an Osgood standard "uniqueness" condition. Nonnegative multi-bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub-and supersolutions that have supports of the same type.

Uniqueness and stability of solutions without the boundary condition

“…Benedikt et al [4,5] had studied the equation u t = div |∇u| p-2 ∇u + q(x) u α-1 u, (x, t) ∈ Q T , with 0 < α < 1, and such that there exists an x 0 ∈ Ω satisfying q(x 0 ) > 0. They showed that the uniqueness of a solution does not hold.…”

On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term

“…al. [8,9] had shown that the uniqueness of the solution of the following equation is true = div (|∇ | −2 ∇ ) + ( ) | | −1 , ( , ) ∈ ( , ) ∈ Ω × (0, ) ,…”

A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations

“…There have been several papers subsequent to [10] providing non-uniqueness results for parabolic equations of various types, e.g. with unbounded coefficients [14], degenerate p-Laplacian operators [3,4,11], and systems [2,6,7,11]. However, all these works either assume explicitly that f is concave near zero [5,9] or implicitly by working only with nonlinearities of power law type, f (u) = u p (0 < p < 1).…”