Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition

Guo, Jong‐Shenq

doi:10.3934/dcds.2007.18.71

Cited by 8 publications

(8 citation statements)

References 13 publications

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“…Others illustrations are given by some plots in the below figures. Figures 1, 2 and 3 show that the numerical solution blows up in finite time on the whole space for γ > 0 and 0 < q < 1, which is in line with the theoretically established result (see [8]).…”

Section: Numerical Experimentssupporting

confidence: 88%

“…Step 2 (Condition A1) From [8], we know that u > 0, u x < 0, u t > 0 and u xx > 0. Let us define the functional J as follows :…”

Section: Numerical Blow-upmentioning

confidence: 99%

“…We know from Jong-Shenq Guo [8] and from Kavitha S., Bhakya K. [11] that, if q > 0 and for every positive bounded smooth initial data u 0 , the solution u of (1.1) blows up in finite time T . Moreover, if q > 1, u 0 ≤ 0 and…”

Section: Introductionmentioning

confidence: 99%

“…The blow-up phenomenon has been the focus of many authors in recent years. Some were interested in the theoretical analysis [5,8,11], and others in the numerical one [1,2,4,10,6,7,12].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Blow-Up on Whole Domain for a Quasilinear Parabolic Equation With Nonlinear Boundary Condition

Ganon¹,

…²,

Taha³

et al. 2020

Adv. Math., Sci. J.

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This paper deals with numerical approximation of the following quasilinear parabolic equationWe show that the solution of the semidiscrete scheme, obtained by the finite differences method blows up in a finite time when 0 < q < 1. Convergence of the numerical blow-up time to the theoretical one when the mesh size goes to zero is also established. Finally, we give some numerical results to illustrate certain point of our work.

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Section: Numerical Experimentssupporting

confidence: 88%

“…Step 2 (Condition A1) From [8], we know that u > 0, u x < 0, u t > 0 and u xx > 0. Let us define the functional J as follows :…”

Section: Numerical Blow-upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The blow-up phenomenon has been the focus of many authors in recent years. Some were interested in the theoretical analysis [5,8,11], and others in the numerical one [1,2,4,10,6,7,12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Blow-Up on Whole Domain for a Quasilinear Parabolic Equation With Nonlinear Boundary Condition

Ganon¹,

…²,

Taha³

et al. 2020

Adv. Math., Sci. J.

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show abstract

“…As for the Neumann or Cauchy problems to the parabolic equations not in divergence forms, one can refer to [2,5,10].…”

Section: Introductionmentioning

confidence: 99%

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

Zhou¹,

Bai²,

Zheng³

2011

Journal of Mathematical Analysis and Applications

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In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: u t = u p u + au q − bu r , subject to the null Dirichlet boundary condition.We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t → ∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.

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Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction‐diffusion equations

Iagar

Sánchez

2023

Math Methods in App Sciences

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Some transformations acting on radially symmetric solutions to the following class of nonhomogeneous reaction‐diffusion equations which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here , , , and , real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self‐similar solutions, which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case .

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Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition

Cited by 8 publications

References 13 publications

Numerical Blow-Up on Whole Domain for a Quasilinear Parabolic Equation With Nonlinear Boundary Condition

Numerical Blow-Up on Whole Domain for a Quasilinear Parabolic Equation With Nonlinear Boundary Condition

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction‐diffusion equations

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