277 research outputs found
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Blow-up analysis for a porous media equation with nonlinear sink and nonlinear boundary condition
In this paper, we study porous media equation ut = ∆u m − u p with nonlinear boundary condition ∂u ∂ν = kuq . We determine some sufficient conditions for the occurrence of finite time blow-up or global existence. Moreover, lower and upper bounds for blow-up time are also derived by using various inequality techniques
Global dynamics of a parabolic type equation arising from the curvature flow
This paper studies a type of degenerate parabolic problem with nonlocal term
\begin{equation*}
\begin{cases}
u_t=u^p(u_{xx}+u-\bar{u}) & 0<t<T_{{\max}},\ 0<x<a,
u_x(0,t)=u_x(a,t)=0 & 0<t<T_{{\max}},
u(x,0)=u_0(x) & 0<x<a,
\end{cases} \end{equation*} where , . In this paper, the
classification of the finite-time blowup/global existence phenomena based on
the associated energy functional and explicit expression of all nonnegative
steady states are demonstrated. Moreover, we combine the applications of
Lojasiewicz-Simon inequality and energy estimates to derive that any bounded
solution with positive initial data converges to some steady state as
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
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