Blow-up of a nonlocal semilinear parabolic equation with positive initial energy

Gao, Wenjie; Han, Yuzhu

doi:10.1016/j.aml.2010.12.040

Cited by 44 publications

(14 citation statements)

References 7 publications

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“…We shall use the improved energy estimate method, which has been successfully applied in [11] to deal with the second-order p-Laplacian equation (see also [12][13][14] for further applications of this method). To extend the method to the sixth-order thin film equation herein, we should combine it with the construction of a second-order elliptic problem (see (18)).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Liu

Chen

2015

Pramana - J Phys

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Section: Proof Of Theoremmentioning

confidence: 99%

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Liu

Chen

2015

Pramana - J Phys

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“…The authors in [7] investigated the blow-up phenomena for solutions of (1.1)-(1.2) with the positive initial energy. They constructed a control function and applied suitable embedding theorems to prove that the classical solutions to the problem (1.1) blow up in finite time provided that E(u 0 ) < E 1 and ∥∇u 0 ∥ 2 > β, where E 1 and β are positive constants which are introduced in their work.…”

Section: Introductionmentioning

confidence: 99%

Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy

Khelghati

Baghaei

2015

Computers & Mathematics with Applications

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a b s t r a c tThis paper is concerned with the blow-up of solutions to the following semilinear parabolic equation:under homogeneous Neumann boundary condition in a bounded domain Ω ⊂ R n , n ≥ 1, with smooth boundary.For all p > 1, we prove that the classical solutions to the above equation blow up in finite time when the initial energy is positive and initial data is suitably large. This result improves a recent result by Gao and Han (2011) which asserts the blow-up of classical solutions for n ≥ 3 provided that 1 < p ≤ n+2 n−2 .

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“…The case when f(u) = u|u| p−1 was studied by [13] and the problem where f(u) = |u| p was considered in [17,30]. In both cases, it was shown that for any p > 1, the solution must blow-up if the initial data satisfies a certain energy condition.…”

Section: Introductionmentioning

confidence: 99%

Transition to blow-up in a reaction–diffusion model with localized spike solutions

2017

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For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

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Blow-up of a nonlocal semilinear parabolic equation with positive initial energy

Cited by 44 publications

References 7 publications

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy

Transition to blow-up in a reaction–diffusion model with localized spike solutions

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