On Blow-Up Solutions of Differential Equations with Poincaré-Type Compactifications

Matsue, Kaname

doi:10.1137/17m1124498

Cited by 29 publications

(95 citation statements)

References 14 publications

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“…Also, the inequality x i ≤ 1 is required for i = 1, · · · , N − 1 for the boundedness of exponential terms h 1,1−x m i ;m (s). In the same argument as [4], we can prove that divergent solutions of (1.1) correspond to global trajectories of (2.4) in {s > 0} asymptotic to equilibria (or general invariant sets) on the horizon {s = 0}. If, moreover, we can prove that maximal existence times t max of calculated solutions shown below are finite, then the corresponding divergent solutions are actually blow-up solutions.…”

Section: We Havementioning

confidence: 80%

“…Typically, compactifications are chosen so that the asymptotically dominant terms at infinity are selected appropriately. Such operations can be done for asymptotically polynomial vector fields and a geometric treatment of blow-up solutions is derived (e.g., [4]). However, the present vector field (1.1) contains exponential nonlinearity, and the general treatment of asymptotic behavior of vector fields at infinity is nontrivial.…”

Section: Desingularized Vector Field At Infinitymentioning

confidence: 99%

“…In this case, the infinity corresponds to the subspace {s = 0} in (x 1 , · · · , x N 2 −1 , s, x N 2 +1,··· ,x N −1 )-coordinate. Following [4], we shall call the subspace {s = 0} the horizon throughout the paper. The infinity (in u N/2component) then corresponds to {s = 0}.…”

Section: Desingularized Vector Field At Infinitymentioning

confidence: 99%

“…The above procedure provides the direct proof of blow-ups for validated solutions in a quantitative way. If we only aim at proving that the validated solution blows up, namely without any concrete data for t max , the asymptotic study around equilibria on the horizon based on arguments in [4] is applied. Using the asymptotic method, we can also calculate the blow-up rate of blow-up solutions (see [8] for details).…”

Section: Finally Inmentioning

confidence: 99%

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Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Matsue

Takayasu

2020

Journal of Computational and Applied Mathematics

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Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of ut = uxx + e u m with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.

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Section: We Havementioning

confidence: 80%

Section: Desingularized Vector Field At Infinitymentioning

confidence: 99%

Section: Desingularized Vector Field At Infinitymentioning

confidence: 99%

Section: Finally Inmentioning

confidence: 99%

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Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Matsue

Takayasu

2020

Journal of Computational and Applied Mathematics

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“…The rest of the paper is organized as follows. In Section 2, we review the approach and result shown in [14,16] about characterization of blow-up solutions in terms of trajectories on stable manifolds of invariant sets at infinity for appropriately associated vector fields. These results gives characterization of type-I blow-ups.…”

mentioning

confidence: 99%

Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

Matsue

2019

Journal of Differential Equations

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Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates.1. Apply compactifications of phase spaces and derivation of desingularized vector fields to (1.3).The infinity then corresponds to a hypersurface E called horizon.2. Specify an invariant set S on E for desingularized vector fields.3. Solve explicit solutions which converge to S. Transform the calculated solutions to those for the original system (1.3).Our main result is the following : if S is non-hyperbolic for desingularized vector field in the sense stated above and we solve trajectories on W c (S), then they correspond to divergent or blowup solutions with blow-up rates which are generally different from type-I. We can also say that the similar feature is revealed to other finite-time singularities such as finite-time extinction, compacton traveling waves or quenching.1 Precise meanings are shown in successive sections. 2The rest of the paper is organized as follows. In Section 2, we review the approach and result shown in [14,16] about characterization of blow-up solutions in terms of trajectories on stable manifolds of invariant sets at infinity for appropriately associated vector fields. These results gives characterization of type-I blow-ups. In Section 3, we discuss a methodology how the asymptotic behavior of blow-up solutions with different blow-up rates from type-I can be derived. The main issue there is a treatment of non-hyperbolic invariant sets at infinity. We thus review the type of equilibria, which is often discussed in singularities of (planar) vector fields (e.g., [7]). Several equivalences among dynamical systems are reviewed, which help us with treating asymptotic behavior of trajectories simply. Independently, we discuss the treatment for detecting other types of finite-time singularities such as finite-time extinctions and compactons. As indicated in preceding studies in partial differential equations (e.g., [12]), there are several common aspects among finite-time singularities including blow-ups, extinctions, compactons and quenching. Our present treatments reveals such common mechanisms of singularities from the viewpoint of dynamical systems. In Section 4, we provide various examples with calculating rigorous rates of finite-time singularities including blow-ups, extinctions, compactons, quenching and periodic blowups. We see that the asymptotic rates can be derived component-wise whether or not it is faster than that associated with nonlinearity of vector fields; namely type-I rates. Compactifications and type-I blow-upsFirst we collect our present sett...

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Numerical validation of blow-up solutions with quasi-homogeneous compactifications

Matsue

Takayasu

2020

Numer. Math.

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On Blow-Up Solutions of Differential Equations with Poincaré-Type Compactifications

Cited by 29 publications

References 14 publications

Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

Numerical validation of blow-up solutions with quasi-homogeneous compactifications

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