Premium
Finite Time Blow‐up for a Non‐linear Parabolic Equation with a Gradient Term and Applications
Author(s) -
Souplet Philippe
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19961110)19:16<1317::aid-mma835>3.0.co;2-m
Subject(s) - mathematics , mathematical analysis , term (time) , a priori and a posteriori , parabolic partial differential equation , simple (philosophy) , population , energy (signal processing) , partial differential equation , physics , statistics , philosophy , demography , epistemology , sociology , quantum mechanics
We give new finite time blow‐up results for the non‐linear parabolic equations u t −Δ u = u p and u t −Δ u +μ∣∇ u ∣ q = u p . We first establish an a priori bound in L p +1 for the positive non‐decreasing global solutions. As a consequence, we prove in particular that for the second equation on ℝ N , with q = 2 p /( p +1) and small μ>0, blow‐up can occur for any N ≥1, p >1, ( N −2) p < N +2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.