Lorentz estimates for asymptotically regular fully nonlinear parabolic equations

We prove a global Lorentz estimate of the Hessian of strong solutions to the Cauchy–Dirichlet problem for a class of fully nonlinear parabolic equations with asymptotically regular nonlinearity over a bounded C^{1, 1} domain. Here, we mainly assume that the associated regular nonlinearity satisfies uniformly parabolicity and the ‐vanishing condition, and the approach of constructing a regular problem by an appropriate transformation is employed.     

Global positivity estimates and Harnack inequalities for the fast diffusion equation

We investigate local and global properties of positive solutions to the fast diffusion equation u_t=Δu^m in the range (d−2)₊/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; we use them to derive sharp global positivity estimates and a global Harnack principle. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak and elliptic Harnack inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.     

On a nonlinear elliptic-parabolic integro-differential equation with L-data

We generalize the concept of entropy solutions for parabolic equations with L¹-data and consider a class of nonlinear history-dependent degenerated elliptic-parabolic equations including problems with a fractional time derivative such as with Dirichlet boundary conditions and initial condition, where 0<γ?1. We show uniqueness of entropy solutions for general L¹-data by Kruzhkov's method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b≡id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.     

A certain class of systems of “reaction-diffusion” type

The Cauchy problem with bounded, nonnegative initial function is investigated for a quasilinear, degenerate, parabolic system of two equations, containing, in general, lower terms with non-power nonlinearities. Their form is such that the first component of the solution can become unbounded after a finite time interval, provided the second component remains strictly positive. However, the latter may tend to zero and even become identically zero after a finite time. Two theorems are proved regarding the existence of a global, bounded generalized solution of the considered problem. Examples are given, attesting to the sharpness of these theorems.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 78–88, 1980.     

On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.     

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation ??u=|?u|^q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation ?Δu?ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onR ⁿ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type?Lu=f(x, u, ?u)+ω where , andL is a uniformly elliptic operator.     

Stabilization of solution to the Cauchy problem for a parabolic equation with lower order coefficients and an exponentially growing initial function

For the coefficients of lower order terms of a second-order parabolic equation, we obtain sharp sufficient conditions under which the solution of the Cauchy problem stabilizes to zero uniformly in x on each compact set K in ? ^N for any exponentially growing initial function.     

Initial trace for a doubly nonlinear parabolic equation

This paper studies the Cauchy problem for a doubly nonlinear parabolic equation. The main result shows that if there is a nonnegative solution of the Cauchy problem, then the initial trace of the solution is uniquely given as a nonnegative Borel measure satisfying an exponential growth condition. This extends the known result for the heat equation to the nonlinear case.     

The Cauchy Problem for Nonlinear Parabolic Equations with Lévy Laplacian

A solution to the Cauchy problem for a rather general class of nonlinear parabolic equations involving the infinite-dimensional Laplacian Δ_L of the form , where f is a real function defined on R³ is presented. Mathematics Subject Classifications (2000) 35R15, 46G05.     

Uniqueness for second-order parabolic equations with discontinuous coefficients

We prove uniqueness of the good solution to the Cauchy–Dirichlet (C–D) problem for linear non-variational parabolic equations with the coefficients of the principal part with discountinuities, in cases in which in general uniqueness of strong solutions in Sobolev spaces does not hold. In particular, we prove uniqueness when the discontinuities of the coefficients are contained in a hyperplane t = t ₀ and, with an extra condition on the eigenvalues of the matrix, in a line segment x = x ₀. Mathematics Subject Classification. 35A05, 35K10, 35K20 Dedicated to the memory of Gene Fabes.     

The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients

We constructed a fundamental matrix of solutions of the Cauchy problem and studied its basic properties for a new class of linear parabolic systems with smooth bounded variable coefficients that includes a class of the Shilov-type parabolic systems of partial differential equations with nonnegative genus.     

Comparison of Green Functions and Harmonic Measures for Parabolic Operators

In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C^1,1-cylindrical domain . Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of . These results are first studied by some authors for certain elliptic and less general parabolic operators. Mathematics Subject Classifications (2000) 31B25, 35B05, 35K10, 58J35.     

Two-sided estimates for solutions to the Cauchy problem for Wazewski linear differential systems with delay

The Cauchy problem is considered for Wazewski linear differential systems with finite delay. The right-hand sides of systems contain nonnegative matrices and diagonal matrices with negative diagonal entries. The initial data are nonnegative functions. The matrices in equations are such that the zero solution is asymptotically stable. Two-sided estimates for solutions to the Cauchy problem are constructed with the use of the method of monotone operators and the properties of nonsingular M-matrices. The estimates from below and above are zero and exponential functions with parameters determined by solutions to some auxiliary inequalities and equations. Some estimates for solutions to several particular problems are constructed.     

Quenching Phenomenon for a Parabolic MEMS Equation

This paper deals with the electrostatic MEMS-device parabolic equation u_t-?u =λf(x)/(1-u)~p in a bounded domain ? of R~N,with Dirichlet boundary condition,an initial condition u0(x) ∈ [0,1) and a nonnegative profile f,where λ 0,p 1.The study is motivated by a simplified micro-electromechanical system(MEMS for short) device model.In this paper,the author first gives an asymptotic behavior of the quenching time T*for the solution u to the parabolic problem with zero initial data.Secondly,the author investigates when the solution u will quench,with general λ,u0(x).Finally,a global existence in the MEMS modeling is shown.     

Supremum principles for the higher-order derivatives of solutions to the Dirichlet problem for parabolic equations without compatibility conditions between the data

The Dirichlet problem is considered for the heat equation ut=auxx, a>0 a constant, for (x,t)∈[0,1]×[0,T], without assuming any compatibility condition between initial and boundary data at the corner points (0,0) and (1,0). Under some smoothness restrictions on the data (stricter than those required by the classical maximum principle), weak and strong supremum and infimum principles are established for the higher-order derivatives, ut and uxx, of the bounded classical solutions. When compatibility conditions of zero order are satisfied (i.e., initial and boundary data coincide at the corner points), these principles allow to estimate the higher-order derivatives of classical solutions uniformly from below and above on the entire domain, except that at the two corner points. When compatibility conditions of the second order are satisfied (i.e., classical solutions belong to on the closed domain), the results of the paper are a direct consequence of the classical maximum and minimum principles applied to the higher-order derivatives. The classical principles for the solutions to the Dirichlet problem with compatibility conditions are generalized to the case of the same problem without any compatibility condition. The Dirichlet problem without compatibility conditions is then considered for general linear one-dimensional parabolic equations. The previous results as well as some new properties of the corresponding Green functions derived here allow to establish uniformL¹-estimates for the higher-order derivatives of the bounded classical solutions to the general problem.     

Hölder continuity of the solutions of a nondiagonal parabolic system of equations with <Emphasis Type="Italic">p</Emphasis>-laplacians

We prove the Hölder continuity of the solutions of the Dirichlet problem for a nondiagonal parabolic system of equations in divergence form whose leading parts are modeled by the p-Laplacians.     

Conjugate Cauchy Problem for Parabolic Shilov Type Systems with Nonnegative Genus

The problem of solvability of the Gelfand–Shilov conjugate Cauchy problem for parabolic Shilov type systems with variable coefficients of bounded smoothness and nonnegative genus in the spaces S _β is studied by constructing the fundamental solution of this problem and analyzing its basic properties.     

On asymptotic proximity of solutions of the Cauchy problem for second-order parabolic equations

A theorem about asymptotic (as t) proximity of weak fundamental solutions of the Cauchy problem is proved for divergent second-order parabolic equations. It is assumed that the coefficients have derivatives generalized in the Sobolev sense. A possible application of this theorem to establishing the uniform proximity of weak solutions of the Cauchy problem is also discussed.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 693–700, May, 1995.     

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Uniqueness classes of solutions of boundary problems for nondivergent second order parabolic equations in noncylindrical domains

In an unbounded noncylindrical domain G we consider classical solutions of general second order linear parabolic equations satisfying a Dirichlet boundary condition on the parabolic part of the boundary of G. On the basis of the maximum principle we single out classes of uniqueness of such solutions depending on the geometry of the domain G which are analogs of the classes of A. N. Tikhonov and S. Täcklind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 924–930, July, 1990.