Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNote on time-regularity for weak solutions to parabolic systems of p-Laplace type
237
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - mbox{div}[mathbb{A}(x,t,u, abla u)]= mbox{div}[{mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(Omega, omega)$-regularity estimates are established for weak solutions of these equations, where $omega$ is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for $omega =1$, because of the singularity of the coefficients in $(x,t)$-variables
We introduce Fundamental solutions of Barenblatt type for the equation $u_t=sum_{i=1}^N bigg( |u_{x_i}|^{p_i-2}u_{x_i} bigg)_{x_i}$, $p_i >2 quad forall i=1,..,N$, on $Sigma_T=mathbb{R}^N times[0,T]$, and we prove their importance for the regularity properties of the solutions.
We study the nonlinear eigenvalue problem for the p-Laplacian, and more general problem constituting the Fucik spectrum. We are interested in some vanishing properties of sign changing solutions to these problems. Our method is applicable in the plane.
We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
