DCDS
Connecting equilibria by blow-up solutions
Marek Fila Hiroshi Matano
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 155-164 doi: 10.3934/dcds.2000.6.155
We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of $L^1$ connections among equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium $\phi^+$ we mean a function $u$($.,t$) which is a classical solution on the interval $(-\infty,T)$ for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a certain weak sense for $t\in [T,\infty)$ and satisfies $u$($.,t$)$\to \phi^\pm$ as $t\to\pm\infty$ in a suitable sense. The main tool in our analysis is the zero number argument; namely to count the number of intersections between the graph of a given solution and that of various specific solutions.
keywords: zero number. connecting orbits blow­up Semilinear parabolic equation nonlinear heat equation
CPAA
Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation
Marek Fila Michael Winkler
Communications on Pure & Applied Analysis 2015, 14(1): 107-119 doi: 10.3934/cpaa.2015.14.107
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast di usion equation with a critical exponent. We improve a previous result on slow convergence to Barenblatt pro les.
keywords: nonlinear Fokker-Planck equation extinction in finite time slow convergence to Barenblatt profiles. Fast diffusion
NHM
Grow up and slow decay in the critical Sobolev case
Marek Fila John R. King
Networks & Heterogeneous Media 2012, 7(4): 661-671 doi: 10.3934/nhm.2012.7.661
We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.
keywords: asymptotic behaviour Sobolev exponent. Semilinear heat equation
DCDS-S
Special asymptotics for a critical fast diffusion equation
Marek Fila Hannes Stuke
Discrete & Continuous Dynamical Systems - S 2014, 7(4): 725-735 doi: 10.3934/dcdss.2014.7.725
We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation $u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.
keywords: Fast diffusion grow-up. extinction in finite time nonlinear Fokker-Planck equation
CPAA
A continuum of extinction rates for the fast diffusion equation
Marek Fila Juan-Luis Vázquez Michael Winkler
Communications on Pure & Applied Analysis 2011, 10(4): 1129-1147 doi: 10.3934/cpaa.2011.10.1129
We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $R^n$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.
keywords: grow-up. extinction in finite time Fast diffusion nonlinear Fokker-Planck equation
CPAA
Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition
Marek Fila Kazuhiro Ishige Tatsuki Kawakami
Communications on Pure & Applied Analysis 2012, 11(3): 1285-1301 doi: 10.3934/cpaa.2012.11.1285
We study the large time behavior of positive solutions for the Laplace equation on the half-space with a nonlinear dynamical boundary condition. We show the convergence to the Poisson kernel in a suitable sense provided initial data are sufficiently small.
keywords: dynamical boundary conditions Poisson kernel. Laplace equation
DCDS
Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems
Marek Fila Hirokazu Ninomiya Juan-Luis Vázquez
Discrete & Continuous Dynamical Systems - A 2006, 14(1): 63-74 doi: 10.3934/dcds.2006.14.63
This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are homogeneous in space blow up in finite time. Is it possible to inhibit the occurrence of blow-up as a consequence of imposing Dirichlet boundary conditions, or other effects where diffusion plays a role? We give examples of equations and systems where the answer is affirmative.
keywords: Blow-up reaction-diffusion Dirichlet conditions prevent blow-up.
DCDS
Convergence to self-similar solutions for a semilinear parabolic equation
Marek Fila Michael Winkler Eiji Yanagida
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 703-716 doi: 10.3934/dcds.2008.21.703
We study the behavior of solutions of the Cauchy problem for a parabolic equation with power nonlinearity. Our concern is the rate of convergence of solutions to forward self-similar solutions. We determine the exact rate of convergence which turns out to depend on the spatial decay rate of initial data.
keywords: Self-similar solutions comparison principle. semilinear parabolic equation critical exponent

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