The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source

• Published in: Nonlinear analysis Vol. 73; no. 7; pp. 2310 - 2323
• Main Authors: Cianci, P., Martynenko, A.V., Tedeev, A.F.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.10.2010; Elsevier
• Subjects: Blow-up solution; Cauchy problem; Degenerate parabolic equation; Estimates; Exact sciences and technology; Existence and nonexistence theorems; Mathematical analysis; Mathematics; Nonlinearity; Partial differential equations; Sciences and techniques of general use; Source term; Variable coefficients; Existence and nonexistence theorems; Blow-up solution; Degenerate parabolic equation; Variable coefficients; Source term; Source terms; Parabolic equation; Existence theorem; Nonlinear analysis; Mathematical model; Degenerate equation
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.06.026

The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form ∂ u ∂ t = d i v ( ρ ( x ) u m − 1 | D u | λ − 1 D u ) + u p is studied. Global in time existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of a solution are obtained in the case of global solvability. A sharp universal (i.e., independent of the initial function) estimate of a solution near the blow-up time is obtained.