Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoundedness of Solutions to a Class of Coercive Systems with Morrey Data
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove global essential boundedness for the weak solutions of divergence form quasilinear systems. The principal part of the differential operator is componentwise coercive and supports controlled growths with respect to the solution and its gradient, while the lower order term exhibits componentwise controlled gradient growth. The x-behaviour of the nonlinearities is governed in terms of Morrey spaces.
rate research
Read More
In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p leq q$; this case is known as $p, q$-growth conditions. Well known counterexamples, even in the simpler case $p=q$, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component $u^alpha$ of the solution $u=(u^1,...,u^m)$ satisfies an improved Caccioppolis inequality and we get the boundedness of $u^{alpha}$ by applying De Giorgis iteration method, provided the two exponents $p$ and $q$ are not too far apart. Let us remark that, in dimension $n=3$ and when $p=q$, our result works for $frac{3}{2} < p < 3$, thus it complements the one of Bjorn whose technique allowed her to deal with $p leq 2$ only. In the final section, we provide applications of our result.
We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Caratheodory maps having Morrey regularity in $x$ and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient.
In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=mathrm{div}(| abla u|^{p-2} abla u)+v^m$, $v_t=mathrm{div}(| abla v|^{q-2} abla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Omegasubset mathbb{R}^N (Ngeq1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable; if $mn=(p-1)(q-1)$ and $Omega$ is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1)^2$, the existence of at least one non-extinction solution for any positive smooth initial data is proved.
Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
In this paper, we consider the initial Neumann boundary value problem for a degenerate kinetic model of Keller--Segel type. The system features a signal-dependent decreasing motility function that vanishes asymptotically, i.e., degeneracies may take place as the concentration of signals tends to infinity. In the present work, we are interested in the boundedness of classical solutions when the motility function satisfies certain decay rate assumptions. Roughly speaking, in the two-dimensional setting, we prove that classical solution is globally bounded if the motility function decreases slower than an exponential speed at high signal concentrations. In higher dimensions, boundedness is obtained when the motility decreases at certain algebraical speed. The proof is based on the comparison methods developed in our previous work cite{FJ19a,FJ19b} together with a modified Alikakos--Moser type iteration. Besides, new estimations involving certain weighted energies are also constructed to establish the boundedness.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
