Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

2008 ◽  
Vol 48 (8) ◽  
pp. 1376-1405 ◽  
Author(s):  
A. V. Gasnikov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Quasilinear Parabolic Equation ◽  
Initial Condition ◽  
The Cauchy Problem ◽  
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.

scholarly journals Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Michael V. Klibanov ◽  
Nikolaj A. Koshev ◽  
Jingzhi Li ◽  
Anatoly G. Yagola
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Weight Function ◽  
Quasilinear Parabolic Equation ◽  
Initial Condition ◽  
Bounded Set ◽  
Ill Posed ◽  
Linear Pdes ◽  
Quasilinear Parabolic

AbstractWe solve numerically the side Cauchy problem for a 1-D parabolic equation. The initial condition is unknown. This is an ill-posed problem. The main difference with previous results is that our equation is quasilinear, whereas known publications on this topic work only with linear PDEs. The key idea is to minimize a weighted Tikhonov functional with the Carleman Weight Function (CWF) in it. Roughly, given a reasonable bounded set of any size in a reasonable Hilbert space, one can choose the parameter of the CWF in such a way that this functional becomes strictly convex on that set.

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