Blowup in diffusion equations: A survey

Bandle, Catherine; Brunner, Hermann

doi:10.1016/s0377-0427(98)00100-9

Cited by 226 publications

(154 citation statements)

References 60 publications

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“…Suppressing the two exponentially growing modes at each of η → ±∞ yields four boundary conditions on this fourth-order linear homogeneous problem, which generically will have only the trivial solution, suggesting that the values of c 0 are isolated. 2 In addition, equation (4.7a) has a conserved integral which can be used to argue that there are only symmetric similarity solutions [10], as is seen in Fig. 7.…”

Section: Similarity Variablesmentioning

confidence: 99%

“…In seminal work, Barenblatt made use of dimensional analysis to classify the types of self-similar solutions observed as intermediate asymptotic states [3,4]. This led to an explosion of work deriving, classifying, computing and rigorously analyzing self-similar solutions in a wide array of systems (see the reviews [3,55,63,2,39,4,40,49,31] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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Section: Similarity Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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“…For p constant we have that there are solutions to (1.1) with T < ∞ if and only if p > 1. In this case, we have lim t T u(·, t) ∞ = +∞, a phenomenon that is called blow-up in the literature and has deserved a great interest, see for instance the books [20], [21], the survey papers [2], [3], [10], [14] and the references therein. However, up to our knowledge, this seems to be the first paper where the blow-up phenomenon is studied with a variable exponent as a reaction term.…”

Section: Introductionmentioning

confidence: 99%

Critical exponents for a semilinear parabolic equation with variable reaction

Ferreira¹,

Pablo²,

Pérez‐Llanos³

et al. 2012

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. In this paper we study the blow-up phenomenon for nonnegative solutions to the following parabolic problem:After discussing existence and uniqueness we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p + > 1.When Ω = R N we show that if p− > 1 + 2/N then there are global nontrivial solutions while if 1 < p − ≤ p + ≤ 1 + 2/N then all solutions to the problem blow up in finite time. Moreover, in case p− < 1+2/N < p+ there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global nontrivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough then the problem possesses global nontrivial solutions regardless the size of p(x).

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“…where (r, φ) are the polar coordinates for D 2 and θ (r, t) satisfies (1.1). PDE (1.6) is the gradient flow associated with the energy = It is known [16,32] that IBVP (1.1)-(1.3) admits a classical solution for a sufficiently smooth initial solution with small energy and a global weak solution for a sufficiently smooth initial solution with finite energy.…”

mentioning

confidence: 99%

“…MMPDEs have been successfully used for the numerical simulation of blowup such as in semilinear parabolic equations [14,15,19,27], Cahn-Hilliard equations [23], and integro-differential equations [31]. In those situations, the solution becomes unbounded and the numerical computation typically stops at a finite time despite the use of a uniform or an adaptive mesh [2,11]. On the other hand, the blowup with (1.1) is different, occurring in the spatial derivative instead of the solution itself.…”

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confidence: 99%

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

Haynes¹

2013

NMTMA

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Abstract. The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.

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Blowup in diffusion equations: A survey

Cited by 226 publications

References 60 publications

Stability and dynamics of self-similarity in evolution equations

Stability and dynamics of self-similarity in evolution equations

Critical exponents for a semilinear parabolic equation with variable reaction

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

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