Symmetrization and Mass Comparison for Degenerate Nonlinear Parabolic
نویسنده
چکیده
We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p-Laplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations. 1991 Mathematics Subject Classification. 35B05, 35B40, 35J60, 35K55, 35K65, 47H20.
منابع مشابه
Symmetrization and Mass Comparison for Degenerate Nonlinear Parabolic and related Elliptic Equations
We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p-Laplacian equation. The results will be used in a companion work in combination with a detailed k...
متن کاملA weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media
In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of the zero-order term, the symmetric rearrangement of the solution u of an elliptic equation, that we write u∗, can be compared pointwise with the so...
متن کاملSymmetrization of exterior parabolic problems and probabilistic interpretation
We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced inBrock andSolynin (TransAmMathSoc 352(4):1759–1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the W...
متن کاملA note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملComparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs
We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang [10], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Sougani...
متن کامل