Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $$\smash {O(t^\frac{1}{2})}$$ O ( t 1 2 ) whereas the support of solutions to the deterministic PME grows only with rate $$\smash {O(t^{\frac{1}{m{+}1}})}$$ O ( t 1 m + 1 ) . The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

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.

Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

When {2N/(N+1)<p<2} and {0<q<p/2}, non-negative solutions to the singular diffusion equation with gradient absorption\partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N}vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition {u_{0}} decays sufficiently rapidly as {|x|\to\infty}. On the one hand, the optimal decay of {u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that {p-1<q<p/2}, optimal extinction rates near the extinction time are derived.