Abstract
This paper considers the singular evolution equation $u_t - \Delta \ln u = 0$, particularly the corresponding Cauchy problem. This equation arises in the study of thin film dynamics and as the formal limit as $m \to 0$ of the porous-medium equation. Through the use of local sup-estimates, similar to those for the porous-medium equation (see E. DiBenedetto and Y. L. Kwong, Intrinsic Harnack estimates and extinction profile for certain parabolic equations, Trans. Amer. Math. Soc., 330 (1992), pp. 783–811), and a global Harnack-type inequality, a critical decay rate for solutions in three or more space dimensions as $|x| \to \infty $ is found. In particular, if the initial data decays faster than this critical rate, there is no solution to the Cauchy problem.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
