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Register a new userIntegral Transform Approach to Generalized Tricomi Equations
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Authors
Karen Yagdjian
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We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
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In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $partial_t^2 u-t^m Delta u=|u|^p$ with initial data $(u(0,cdot), partial_t u(0,cdot))= (u_0, u_1)$, where $tgeq 0$, $xin{mathbb R}^n$ ($nge 3$), $minmathbb N$, $p>1$, and $u_iin C_0^{infty}({mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{text{crit}}(m,n)>1$ such that the solution $u$, in general, blows up in finite time when $1<p<p_{text{crit}}(m,n)$. We further show that there exists a conformal exponent $p_{text{conf}}(m,n)> p_{text{crit}}(m,n)$ such that the solution $u$ exists globally when $p>p_{text{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{text{crit}}(m,n)<pleq p_{text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.
In this paper, we consider the blow-up problem of semilinear generalized Tricomi equation. Two blow-up results with lifespan upper bound are obtained under subcritical and critical Strauss type exponent. In the subcritical case, the proof is based on the test function method and the iteration argument. In the critical case, an iteration procedure with the slicing method is employed. This approach has been successfully applied to the critical case of semilinear wave equation with perturbed Laplacian or the damped wave equation of scattering damping case. The present work gives its application to the generalized Tricomi equation.
In the present paper, we study the Cauchy problem for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds. Our aim of this paper is to prove a small data blow-up result and an upper estimate of lifespan of the problem for a suitable compactly supported initial data in the subcritical and critical cases of the Strauss type. The proof is based on the framework of the argument in the paper [17]. One of our new contributions is to construct two families of special solutions to the free equation (see (2.16) or (2.18) as the test functions and prove their several properties. We emphasize that the system with two different propagation speeds is treated in this paper and the assumption on the initial data is improved from the point-wise positivity to the integral positivity.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.
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