The author proves blow up of solutions to the cauchy problems of certain nonlinear wave equations and, also, estimates the time when the blow up occurs. Cauchy problem au 0 in a neighborhood of t with data u yl\ xj2 and gradw xi, xi. In this session we solve cauchy problems for wave equations. Jim lambers mat 606 spring semester 201516 lecture 12 and notes these notes correspond to section 4. As suggested by our terminology, the wave equation 1.
By analogy with the cauchy problem for second order o. Wave equation cauchy problem cauchy data cylindrical wave progressive wave these keywords were added by machine and not by the authors. Cauchy problems for wave equations updated to maple 7. Burq and tzvetkov 10 obtained the global existence result of the cauchy problem for a supercritical wave equation. Most of you have seen the derivation of the 1d wave equation from newtons and hookes.
In 3 we present the ist for the 2ddt equation 12 and use it to obtain the formal solution of the cauchy for such equation. A cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. Without loss of generality, we assume fx gx 0, because we can always add the solution of this problem to a solution of the homogeneous wave equation to obtain a solution of the nonhomogeneous problem with general initial data. The initial value problem cauchy problem for the 1dimenisonal wave equation is given by utt. In a recent paper2 riemanns method for the solution of the problem of cauchy for a linear hyperbolic partial differential equation lu 0 of second order for one unknown function. Vilnius university, faculty of mathematics and informatics, naugarduko 24, vilnius, lithuania. Intuitive interpretation of the wave equation the wave equation states that the acceleration of the string is proportional to the tension in the string, which is given by its concavity.
The global cauchy problem for the non linear schrodinger. Elliptic cauchy problem for one parameter family of domains 4 3. A typical hyperbolic equation is the wave equation. Thus a cauchy problem may have more than one solution. Pdf the global cauchy problem for the critical nonlinear. Cauchy problems advanced engineering mathematics 5 7. Solve the following initial value problem for the cauchyeuler equation. The mathematics of pdes and the wave equation mathtube. On the cauchy problem for the wave equation on timedependent domains gianni dal maso and rodica toader abstract. Uniqueness theorem of the cauchy problem for schrodingers equation in weighted sobolev spaces dan, yuya, differential and integral equations, 2005. The cauchy problem for the wave equation using the.
Shapiro and khavinson conjecture that any solution of the laplace equation with cauchy data given on an analytic surface f can be analytically continued in r or c as far as the schwarz potential of t can be. We prove an existence result for the solution of the cauchy problem and present some. The cauchy problem for the wave equation in a bounded domain let the cauchy problem 1 and 2 be formulated for. But it is often more convenient to use the socalled dalembert solution to the wave equation 1. Stability of the cauchy problem for wave equations 5. Some exact solutions of a heat wave type of a nonlinear. Cauchy boundary conditions are analogous to the initial conditions for a. Global solution to the cauchy problem of the nonlinear. Random data cauchy problem for the wave equation on. For derivation of the equation and physical meaning of boundary conditions, check salsas book. Modeling the longitudinal and torsional vibration of a rod, or of sound waves. On the cauchy problem for the generalized shallow water wave equation article pdf available in journal of differential equations 2457. Chapter 7 heat equation partial differential equation for temperature ux,t in a heat conducting insulated rod along the xaxis is given by the heat equation.
Secondorder hyperbolic partial differential equations wave equation linear wave equation. In this paper, some known and novel properties of the cauchy and signaling problems for the onedimensional timefractional diffusion wave equation with the caputo fractional derivative of order. Consider the cauchy problem for the quasilinear equation in two variables a. The selection of permissable wavenumbers k that apply in a particular problem will be determined by solving the appropriate eigenvalue problem. The solution to the cauchy problem with ux,01 for all x is ux,t1 for all x and t 0. Cauchy problem for a first order quasi linear pde youtube. The hyperbolic equations constitute a broad class of equations for which the cauchy problem is wellposed.
We have solved the wave equation by using fourier series. However vx,t 0 is also a solution to the same cauchy problem. Chapter 6 partial di erential equations most di erential equations of physics involve quantities depending on both. Pdf on the cauchy problem for the generalized shallow water. Pdf blow up of solutions to the cauchy problem for. Second order homogeneous cauchy euler equations consider the homogeneous differential equation of the form. The solution to the initial value problem is ux,t e. Since we assume that the bicharacteristic curve is not locally contained in w f u, there is a point. In 4 we obtain the longtime behaviour of the solutions of such a cauchy problem, showing that the solutions break. Every solution of the wave equation utt c2uxx has the form ux, t fx. We study the inverse cauchy problem for a time fractional di usion wave equation with distributions in righthand sides.
Singbal no part of this book may be reproduced in any form by print, micro. A differential equation in this form is known as a cauchy euler equation. Consider the cauchy problem for the wave equation in rn, namely. The cauchy problem for the wave equation in a bounded domain let the cauchy problem 1 and 2 be formulated for x e sz c r bounded, and let us assume, for simplicity of presentation, that s2 is the parallelepiped 52 0, al x 0, a2l x. Global solution to the cauchy problem of the nonlinear double dispersive wave equation with strong damping c. The cauchy problem and wave equations springerlink. If a thermal conductivity does not depend on temperature, we have the linear equation. We shall consider wellposed problems for the wave equation in two and three variables. Our plan is to identify the real and imaginary parts of f, and then check if the cauchy riemann equations hold for them.
It is like having p0x 0 in an ordinary di erential equation. Bourgain and bulut 46 studied gibbs measure evolution in radial nonlinear wave on a three dimensional ball. The cauchy problem for partial differential equations of. In this case the cauchy problem is global in nature, but the condition that be noncharacteristic is no longer sufficient. We now verify that this solution formula indeed yields a solution of the nonhomogeneous wave equation. The wave equation in a general spherically symmetric black hole geometry masarik, matthew, advances in theoretical and mathematical physics, 2011. On the global cauchy problem for the nonlinear schrodinger. The problem is to nd a solution continuous in time in generalized sense of the direct problem and an unknown continuous timedependent part of a source. Introduction to partial di erential equations, math 4635.
Eigenvalues of the laplacian laplace 323 27 problems. Srivastava, on the occasion of his 65th birth anniversary abstract the paper is devoted to the study of the cauchy type problem for the di. While this solution can be derived using fourier series as well, it is. This video shows how to deal with cauchy problem for inhomogeneous second order differential equation with constant coefficients. Project supported by the national natural science foundation of china and a doctoral program and a key teachers program of the natural science foundation of the ministry of education of china. Nov 26, 2002 on the global cauchy problem for the nonlinear schrodinger equation jean bourgain proceedings of the national academy of sciences nov 2002, 99 24 1526215268. In this chapter, we prove that cauchy problem for wave equation is. The scheme of generating radial solutions to cauchy problems as limits of exterior. Since we assumed k to be constant, it also means that material properties. When c 2 the wave forms are bellshaped curves moving to the right at speed 2.
For the wave equation in r3 the di erent convergence behavior of exterior neumann and exterior dirichlet solutions is brought out by considering h2 vs. This method consists of inferring the solution of the problem referred to from the well known solution of the same problem for one space variable. Burq and tzvetkov 11 established the probabilistic wellposedness for 1. Cauchy data 197 if this expression is zero, we are stuck. Lectures on cauchy problem by sigeru mizohata notes by m. Random data cauchy problem for the wave equation on compact. A cauchy problem can be an initial value problem or a boundary value problem for this case see also cauchy boundary condition or it can be either of them. Wellposedness of cauchy problem in this chapter, we prove that cauchy problem for wave equation is wellposed see appendix a for a detailed account of wellposedness by proving the existence of a solution by explicitly exhibiting a formula, followed by uniqueness of solutions to cauchy problem. The cauchy problem for wave equations with non lipschitz coefficients ferruccio colombini. The unique solvability of the problem is established.
Cauchy type problem for diffusion wave equation with the riemannliouville partial derivative anatoly a. E r c r bounded, and let us assume, for simplicity of presentation, that fl is the parallelepiped r 0, ui x 0, a2 x. If a function f is analytic on a simply connected domain d and c is a simple closed contour lying in d then. The maximum difference between the initial functions is 1n which gets smaller and smaller as n grows. The cauchy problem for elliptic equations 29 references 40 1. Now let us find the general solution of a cauchy euler equation. Cauchy problem of semilinear inhomogeneous elliptic. The cauchy problem for the nonhomogeneous wave equation. Separation of variables wave equation 305 25 problems.
In this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form. Levine and grozdena todorova communicated by david s. Integral surface passing through a given curve 1 cauchy problem for a first order quasi linear pde. The literature on semilinear wave equations is vast, yet we have complete existence results for only some special cases of semilinearities. Applications other applications of the onedimensional wave equation are. Separation of variables heat equation 309 26 problems. In the preceding examples, we note that f x is continuous, but not. Homogeneous euler cauchy equation can be transformed to linear constant coe cient homogeneous equation by changing the independent variable to t lnx for x0. Consider the cauchy problem for the quasilinear equation in two variables. A large amount of work has been devoted in the last few years to the study of the cauchy problem for the nonlinear schrodinger equation 1. The cauchy problem for wave equations with non lipschitz.
A method of ascent is used to solve the cauchy problem for linear partial differential equations of the second order in p space variables with constant coefficients i. An important result about cauchy problems for ordinary differential equations is the existence and uniqueness theorem, which states that, under mild assumptions, a cauchy problem always admits a unique solution in a neighbourhood of the. This process is experimental and the keywords may be updated as the learning algorithm improves. The question of the well posedness of the cauchy problem for the wave equation with nonsmooth coe cients has already been studied in the case that the second order part has the special form, in coordinates.
Pdf the author proves blow up of solutions to the cauchy problem of certain nonlinear wave equations and, also, estimates the time when the blow up. Thus, our initial or cauchy data are ux,0 gx,uhx,x. Eigenvalues of the laplacian poisson 333 28 problems. The cauchy problem for differential equations a guide to.
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