Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTime regularity and long-time behavior of parabolic $p$-Laplace equations on infinite graphs
122
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider the so-called emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator of the associated nonlinear semigroups. We prove higher order time regularity of the solutions. We investigate the long-time behavior of the solutions and discuss in particular finite extinction time and conservation of mass. Namely, on one hand, for small $p$ if an infinite graph satisfies some isoperimetric inequality, then the solution to the parabolic $p$-Laplace equation vanishes in finite time; on the other hand, for large $p,$ these parabolic $p$-Laplace equations always enjoy conservation of mass.
rate research
Read More
We show that local weak solutions to parabolic systems of p-Laplace type are H{o}lder continuous in time with values in a spatial Lebesgue space and H{o}lder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee-Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions.
We seek solutions $uinR^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newtons method, create informative graphics, and be in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithms capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.
We study short--time existence, long--time existence, finite speed of propagation, and finite--time blow--up of nonnegative solutions for long-wave unstable thin film equations $h_t = -a_0(h^n h_{xxx})_x - a_1(h^m h_x)_x$ with $n>0$, $a_0 > 0$, and $a_1 >0$. The existence and finite speed of propagation results extend those of [Comm Pure Appl Math 51:625--661, 1998]. For $0<n<2$ we prove the existence of a nonnegative, compactly--supported, strong solution on the line that blows up in finite time. The construction requires that the initial data be nonnegative, compactly supported in $R^1$, be in $H^1(R^1)$, and have negative energy. The blow-up is proven for a large range of $(n,m)$ exponents and extends the results of [Indiana Univ Math J 49:1323--1366, 2000].
We establish quantitative results for the statistical be-ha-vi-our of emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,mu)$ preserving a $sigma$-finite measure $mu$ such that $mu(X)=infty$; ii) the case where $mu$ is a probability measure but we consider the statistical behaviour of an observable $phicolon Xto[0,infty)$ which is non-integrable: $int phi , dmu=infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove asymptotic relations between the behaviour of $phi $, the local dimension of $mu$, and on the growth of Birkhoff sums (as time tends to infinity). We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel--Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
