Generalization of the Lax–Ryabenky–Philippov theorem to nonlinear problems

In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h0. In this case, the value of h0 depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.

Extinction in a finite time for solutions of a class of quasilinear parabolic equations

We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t ( u q ) − ∇ ( | ∇ u | p − 2 ∇ u ) + a ( x ) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0, p ⩾ 1 + q, p ⩾ 2, a ( x ) ⩾ 0 and Ω a bounded domain of R N ( N ⩾ 1). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a ( x ) when p = 1 + q.

Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

Exponential decay for quasilinear parabolic equations in any dimension

<p style='text-indent:20px;'>We estimate decay rates of solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in any dimension. Such decay rates depend only on the constitutive relations, spatial domain, and range of the initial function.</p>