Introduction.- 1. Global solutions to wave equations - existence theorems.- 2. L^p - L^q-decay estimates for the linear we equation.- 3. Linear symmetric hyperbolic systems.- 3.1 Energy estimates.- 3.2 A global existence theorem.- 3.3 Remarks on other methods.- 4. Some inequalities.- 5. Local existence for quasilinear symmetric hyperbolic.- 6. High energy estimates.- 7. Weighted a priori estimates.- 8. Global solutions to wave equations - proofs.- 8.1 Proof of Theorem 1.1.- 8.2 Proof ot Theorem 1.2.- 9. Other methods.- 10. Development of singularities.- 11. More evolutions equations.- 11.1 Equations of elasiticity.- 11.1.1 Initially isotropic media in R^3.- 11.1.2 Initially cubic media in R^3.- 11.2 Heat equations.- 11.3 Equations of thermoelasticity.- 11.4 Schrodinger equations.- 11.5 Klein-Gordon equations.- 11.6 Maxwell equations.- 11.7 Plate equations.- 12. Further aspects and questions.- 13. Evolution equations in waveguides.- 13.1 Nonlinear wave equations.- 13.1.1 Linear part.- 13.1.2 Nonlinear part.- 13.2. Schrodinger equations.- 13.3. Equations of elasticity and Maxwell equations.- 13.4 General waveguides.- Appendix.- A. Interpolation.- B. The Theorem of Cauchy-Kowalevsky.- C. A local existence theorem for hyperbolic-parabolic systems References Notation Index.

Lectures on nonlinear evolution equations

In recent years, the theory of Borel summability or multisummability of divergent
					power series of one variable has been established and it has been proved that
					every formal solution of an ordinary differential equation with irregular
					singular point is multisummable. For partial differential equations the
					summability problem for divergent solutions has not been studied so well, and in
					this paper we shall try to develop the Borel summability of divergent solutions
					of the Cauchy problem of the complex heat equation, since the heat equation is a
					typical and an important equation where we meet diveregent solutions. In
					conclusion, the Borel summability of a formal solution is characterized by an
					analytic continuation property together with its growth condition of Cauchy data
					to infinity along a stripe domain, and the Borel sum is nothing but the solution
					given by the integral expression by the heat kernel. We also give new ways to
					get the heat kernel from the Borel sum by taking a special Cauchy data.

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On the Borel summability of divergent solutions of the heat equation

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schr\"odinger equation and other equations.

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The Mathematics of Superoscillations

Multisummability of formal power series solutions of partial differential equations with constant coefficients

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

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Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations. In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

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Newton polygons and Gevrey indices for linear partial differential operators

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x, u,Dxu) = 0 with u(0) = 0. Here the function f(x, u, ξ) is defined and holomorphic in a neighbourhood of a point (0, 0, ξ0) ∈ Cx × Cu × Cξ (ξ 0 = Dxu(0)) and f(0, 0, ξ0) = 0. The equation (SE) is said to be singular if f(0, 0, ξ) ≡ 0 (ξ ∈ C). The criterion of convergence of a formal solution u(x) = ∑ |α|≥1 uαx α of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal solution diverges a precise rate of divergence or the formal Gevrey order is specified which can be interpreted in terms of the Newton polygon as in the case of linear equations but for nonlinear equations it depends on the individual formal solution. 0. Introduction. We write x = (x1, . . . , xn) ∈ C (n ≥ 1), u ∈ C and ξ = (ξ1, . . . , ξn) ∈ C for the complex variables. Let f(x, u, ξ) be a holomorphic function in a neighbourhood of the origin of the cited variables, and assume f to be an entire function in ξ ∈ C when x, u are fixed. In this paper we are concerned with formal solutions of the first order nonlinear partial differential equation (SE) f(x, u(x), Dxu(x)) = 0, u(0) = 0, where Dxu = (D1u, . . . ,Dnu) (Dj = ∂/∂xj). We always assume that the equation (SE) is singular in the sense that (S) f(0, 0, ξ) ≡ 0, ξ ∈ C. 2000 Mathematics Subject Classification: 35A10, 35C10, 35F20.

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Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=15&iss=2&rank=2

Masatake Miyake

Papers

On the Borel summability of divergent solutions of the heat equation

Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations

Newton polygons and Gevrey indices for linear partial differential operators

Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain

On Cauchy-Kowalevski's Theorem for General Systems