On the Cauchy problem for a very fast diffusion equation

Journal Article · Tue Dec 31 04:00:00 EST 1996 · Communications in Partial Differential Equations

OSTI ID:494314

Guo, Jong-Shenq ^[1]

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1. National Taiwan Normal Univ. (China)

In this paper, we study the Cauchy problem for the following nonlinear parabolic equation u{sub t} = {Delta}(ln u) in R{sup N} x (0, {infinity}) with the initial condition u(x, 0) = u{sub 0}(x), x {element_of} R{sup N}, where N {ge} 2 and u{sub 0} is a nonnegative locally integrable function in R{sup N}. Note that the equation (1.1) is an equation of the type u{sub t} = {triangledown} {center_dot} (u{sup m-1}{triangledown}u) with m = 0. the equation (1.3) is a slow diffusion equation if m > 1; the standard heat equation if m = 1; a fast diffusion equation if 0 < m < 1; and a very fast diffusion equation if m {le} 0. Equation (1.3) and its applications have been studied very extensively for past years, see, for example, the survey papers for m > 0, and some partial list for m {le} 0. 31 refs.

• OSTI ID: 494314
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 9-10 Vol. 21; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
CAUCHY PROBLEM
DIFFUSION
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS