Analytic solutions of some nonlinear diffusion equations

Masuda, Koichi

doi:10.1007/bf01163166

Cited by 17 publications

(18 citation statements)

References 11 publications

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“…Motivated by analyticity, Stuke studied extensions to complex, rather than real, time t. Of course this entails complex u. Following an idea of Masuda [53] and Guo at al. [33], Stuke attempts to bypass real time blow-up at t = T by an excursion into the complex plane; see Fig.…”

Section: Dynamics At Infinity and Blow-up In Complex Timementioning

confidence: 99%

Global Dynamics, Blow-Up, and Bianchi Cosmology

Ben-Gal,

Brehm,

Buchner

et al. 2016

Preprint

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Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.We survey some of our results supported in the framework of the Sonderforschungsbereich 647 "Space -Time -Matter" of the Deutsche Forschungsgemeinschaft.

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Section: Dynamics At Infinity and Blow-up In Complex Timementioning

confidence: 99%

Global Dynamics, Blow-Up, and Bianchi Cosmology

Ben-Gal,

Brehm,

Buchner

et al. 2016

Preprint

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“…Furthermore this equation arises from the mathematical studies by Masuda [Mas83,Mas84] and numerical observations by Cho et al [COS16] to consider the nonlinear heat equation in the complex plane of time…”

Section: Introductionmentioning

confidence: 99%

“…As an innovative work, Masuda has considered the Cauchy problem of (2) under Neumann boundary conditions in [Mas83,Mas84]. He has proved global well-posedness of analytic solutions of (2) in a specific domain and existence of branching singularities at the movable singularity, which is called blow-up times in the case of real-valued nonlinear heat equations.…”

Section: Introductionmentioning

confidence: 99%

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

Jaquette,

Lessard,

Takayasu

2021

Preprint

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In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computerassisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.

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“…Here, the subscript z denotes the complex-derivative with respect to z and Re/Im denote the real/imaginary part of complex values, respectively. In his pioneering works [11,12] for this problem, Masuda considered the solution of (1.2) under the Neumann boundary condition and proved global existence of the solution in the shaded domain of Fig. 1 (a) if the initial data u 0 (x) is close to a constant.…”

Section: Introductionmentioning

confidence: 99%

Rigorous numerics for nonlinear heat equations in the complex plane of time

Takayasu¹,

Lessard²,

Jaquette³

et al. 2019

Preprint

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In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov-Perron method to calculate part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

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Analytic solutions of some nonlinear diffusion equations

Cited by 17 publications

References 11 publications

Global Dynamics, Blow-Up, and Bianchi Cosmology

Global Dynamics, Blow-Up, and Bianchi Cosmology

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

Rigorous numerics for nonlinear heat equations in the complex plane of time

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