Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincar\'e inequality, established in [38] for stratified groups

Blow-up phenomena for a pseudo-parabolic system with variable exponents

In this paper, we consider a pseudo-parabolic system with nonlinearities of variable exponent type \begin{align*} \begin{cases} u_t-\Delta u_t-\operatorname{div}(|\nabla u|^{m(x)-2}\nabla u)=|uv|^{p(x)-2}uv^2 & \mbox{in}\ \Omega\times(0,T),\\ v_t-\Delta v_t-\operatorname{div}(|\nabla v|^{n(x)-2}\nabla v)=|uv|^{p(x)-2}u^2v & \mbox{in}\ \Omega\times(0,T) \end{cases} \end{align*} associated with initial and Dirichlet boundary conditions, where the variable exponents $p(\cdot)$, $m(\cdot)$, $n(\cdot)$ are continuous functions on $\overline{\Omega}$. We obtain an upper bound and a lower bound for blow-up time if variable exponents $p(\cdot)$, $m(\cdot)$, $n(\cdot)$ and the initial data satisfy some conditions

Blow-up phenomena for a p(x)-biharmonic heat equation with variable exponent

In this paper, we deal with a p(x)-biharmonic heat equation with variable exponent under Dirichlet boundary and initial condition. We prove the blow up of solutions under suitable conditions

Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II