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Register a new userOn the Local Behavior of Local Weak Solutions to some Singular Anisotropic Elliptic Equations
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We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior Holder continuity, and some integral and pointwise Harnack inequalities.
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We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
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 $La(v^m/m)+alpha_m v+beta xcdot abla v=0$, $v>0$, 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 $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$.
We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musielas stochastic PDE.
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