اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPointwise estimates for solutions of singular quasi-linear parabolic equations
181
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
For a class of singular divergence type quasi-linear parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via the nonlinear Wolff potentials.
قيم البحث
اقرأ أيضاً
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
For a class of divergence type quasi-linear degenerate parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via nonlinear Wolff potentials.
For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for
the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i
.e., the solution of $Delta_{p}u=0, xinOmega$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Delta_{p}u=|u|^{lambda-1}u, xinOmega$, which are then generalized for the equation with general right hand term $f(x,u)$, under some asymptotic conditions of $f$. Lastly, we get pointwise estimates for higher order derivatives of the solution of $-Delta u=u^{lambda},xinOmega$, the case of $p=2$ for p-Laplace equation.
Let $ngeq 3$, $0le m<frac{n-2}{n}$, $rho_1>0$, $beta>beta_0^{(m)}=frac{mrho_1}{n-2-nm}$, $alpha_m=frac{2beta+rho_1}{1-m}$ and $alpha=2beta+rho_1$. For any $lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $La(v^m/m)+alph
a_m v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$ which satisfies $lim_{|x|to 0}|x|^{frac{alpha_m}{beta}}v^{(m)}(x)=lambda^{-frac{rho_1}{(1-m)beta}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $mto 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $R^nsetminus{0}$ to the solution $v$ of $Lalog v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nbs{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$. We also prove that if the solution $u^{(m)}$ of $u_t=Delta (u^m/m)$, $u>0$, in $(R^nsetminus{0})times (0,T)$ which blows up near ${0}times (0,T)$ at the rate $|x|^{-frac{alpha_m}{beta}}$ satisfies some mild growth condition on $(R^nsetminus{0})times (0,T)$, then as $mto 0^+$, $u^{(m)}$ converges uniformly in $C^{2+theta,1+frac{theta}{2}}(K)$ for some constant $thetain (0,1)$ and any compact subset $K$ of $(R^nsetminus{0})times (0,T)$ to the solution of $u_t=Lalog u$, $u>0$, in $(R^nsetminus{0})times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $La log v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
