On a class of degenerate parabolic equations of Kolmogorov type
نویسنده
چکیده
We adapt the Levi’s parametrix method to prove existence, estimates and qualitative properties of a global fundamental solution to ultraparabolic partial differential equations of Kolmogorov type. Existence and uniqueness results for the Cauchy problem are also proved.
منابع مشابه
Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equation...
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 کاملHarnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
متن کاملGradient estimates for degenerate quasi-linear parabolic equations
For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients infinitesimally form bounded with respect to the Laplacian we obtain Lq-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 var...
متن کاملDissipative and Entropy Solutions to Non-isotropic Degenerate Parabolic Balance Laws
Abstract. We propose a new notion of weak solutions (dissipative solutions) for nonisotropic, degenerate, second order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic,...
متن کامل