NUMERICAL QUENCHING FOR A NONLINEAR DIFFUSION EQUATION WITH SINGULAR BOUNDARY FLUX

In this paper, we study the semidiscrete approximation of the solution of a nonlinear diffusion equation with nonlinear source and singular boundary flux. We find some conditions under which the solution of the semidiscrete form quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.

Similarity solutions for early-time constant boundary flux imbibition in foams and soils

The foam drainage equation and Richards equation are transport equations for foams and soils, respectively. Each reduces to a nonlinear diffusion equation in the early stage of infiltration during which time, flow is predominantly capillary driven, hence is effectively capillary imbibition. Indeed such equations arise quite generally during imbibition processes in porous media. New early-time solutions based on the van Genuchten relative diffusivity function for soils are found and compared with the same for drainage in foams. The moisture profiles which develop when delivering a known flux into these various porous materials are sought. Solutions are found using the principle of self-similarity. Singular profiles that terminate abruptly are obtained for soils, a contrast with solutions obtained for node-dominated foam drainage which are known from the literature (the governing equation being now linear is analogous to the linear equation for heat transfer). As time evolves, the moisture that develops at the top boundary when a known flux is delivered is greater in soils than in foams and is greater still in loamy soils than in sandstones. Similarities and differences between the various solutions for nonlinear and linear diffusion are highlighted. Graphic abstract

Asymptotic properties of quenching for the quasilinear electrostatic micro-electro-mechanical system

This study focuses on the asymptotic properties of quenching solutions for the quasilinear parabolic electrostatic micro-electro-mechanical system with singular logarithm boundary flux. Under some assumptions on initial data, quenching in finite time, blowup phenomenon of a time derivative, lower and upper bounds of quenching rate are studied theoretically, and the asymptotic behaviors near quenching time are verified numerically.

Blow-up analysis for a reaction-diffusion equation with gradient absorption terms

<abstract><p>This paper deals with the blow-up phenomena of solution to a reaction-diffusion equation with gradient absorption terms under nonlinear boundary flux. Based on the technique of modified differential inequality and comparison principle, we establish some conditions on nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, some bounds for blow-up time are derived under appropriate measure in higher dimensional spaces $ \left({N \ge 2} \right). $</p></abstract>