Qi S. Zhang

A blow-up result for a nonlinear wave equation with damping: The critical case

Abstract Using a different and much shorter approach, we prove a blow-up result which is more general than the interesting blow-up result of G. Todorova and B. Yordanov concerning a nonlinear wave equation with a damping term. We also show that the critical exponent belongs to the blow-up case. This problem had been left open by these authors.

SHARP GRADIENT ESTIMATE AND YAU'S LIOUVILLE THEOREM FOR THE HEAT EQUATION ON NONCOMPACT MANIFOLDS

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamiltons estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yaus celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in

Some gradient estimates for the heat equation on domains and for an equation by Perelman

{\mathbb R}^n

Blow-up results for nonlinear parabolic equations on manifolds

without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.

Global solutions of inhomogeneous Hamilton-Jacobi equations

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

Linear parabolic equations with strong singular potentials

1. Introduction. The aim of this paper is threefold. First, by a unified approach, we prove that several classical blow-up results obtained over the last three decades for semilinear and quasilinear parabolic problems in R n are valid on noncompact, complete Riemannian manifolds, which include those with nonnegative Ricci curvatures. Next, we remove some unnecessary a priori growth conditions on solutions of the quasilinear case, which are assumed in the existing literature. Finally, we demonstrate a new critical phenomenon for some inhomogeneous, quasilinear parabolic equations. We also hope that this paper serves as a link for the many other papers on this subject, which lie scattered in several journals over a period of three decades. Specifically, we study the blow-up properties of the following homogeneous and inhomogeneous, semilinear parabolic equations and of the porous medium equations with nonlinear source: u − ∂ t u + V (x)u p = 0 in M n × (0, ∞), u(x, 0) = u 0 (x) in M n , u ≥ 0, (1.1)

On a parabolic equation with a singular lower order term

We consider the viscous Hamilton-Jacobi (VHJ) equationut-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas considerable effort has been devoted to study the large time behavior of solutions of the equationut-Δu=g(x,u), whereamplitude blow-up may occur if for instanceg(x,u)≈up asu→∞ andp>1, relatively little is known in the case of (VHJ). The aim of this paper is to investigate this question. More precisely, we study the relations between(i)the existence of global classical solutions(ii)the existence of stationary solutions (with gradient possibly singular on the boundary);and we obtain a precise description of the global dynamics for (VHJ). Namely, we show that (i) implies (ii) and that in this case, all global solutions converge uniformly to the (unique) stationary solution. In the radial case, we prove that, conversely, (ii) implies (i). Moreover, for certain (smooth) functionsh, we obtain the existence of global classical solutions with gradient blowing up in infinite time. For 1p-2 or for the Cauchy problem, all solutions are global, but we establish similar relations between the existence of bounded or locally bounded solutions and the existence of stationary solutions. Our proofs depend on some new gradient estimates of solutions, local and global in space, obtained by Bernstein type arguments. As another consequence of these estimates we prove a parabolic Liouville-type theorem for solutions ofut\t-Δu=│Δu│p in ℝNx(\t-\t8,0). Various other results are obtained, including universal bounds for global solutions.

Finite-Time Blowup for Wave Equations with a Potential

Using an extension of a recent method of Cabre and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabre and Martel from bounded smooth domains to unbounded nonsmooth domains.

The global behavior of heat kernels in exterior domains

Abstract. We obtain the existence of the weak Green’s functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green’s function, we prove a Harnack inequality for the non-negative weak solutions.

A Harnack inequality for the equation ∇(a∇u)+b∇u=0, when |b|∈Kn+1

First we give a truly short proof of the major blowup result [T. C. Sideris, J. Differential Equations, 52 (1984), pp. 378--406] on higher-dimensional semilinear wave equations. Using this new method, we also establish blowup phenomenon for wave equations with a potential. This complements the recent interesting existence result by [V. Georgiev, C. Heiming, and H. Kubo, Comm. Partial Differential Equations, 26 (2001), pp. 2267--2303], where the blowup problem was left open.