The instability of stationary inhomogeneous solutions of the Cauchy problem for the quasilinear parabolic equation and its ecological applications

1980 ◽  
Vol 20 (5) ◽  
pp. 235-241
Author(s):  
V.A. Razzhevaikin
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Quasilinear Parabolic Equation ◽  
The Cauchy Problem ◽  
Ecological Applications ◽  
Quasilinear Parabolic

Blow-up of solutions of a quasilinear parabolic equation

2012 ◽  
Vol 142 (2) ◽  
pp. 425-448
Author(s):  
Ryuichi Suzuki ◽  
Noriaki Umeda
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Positive Function ◽  
Quasilinear Parabolic Equations ◽  
The Cauchy Problem ◽  
Image Position ◽  
Quasilinear Parabolic

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = Δum + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up conditionWe show that if ξm+2/N /(−log ξ)β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limtt→∞ inf|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.

HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA

2006 ◽  
Vol 08 (03) ◽  
pp. 331-354 ◽  
Author(s):  
V. A. GALAKTIONOV ◽  
A. E. SHISHKOV
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Quasilinear Parabolic Equation ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Semilinear Heat Equation ◽  
Basic Model ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Quasilinear Parabolic

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,p is the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN × R+ with arbitrary singular initial data u0 ≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

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