Uniqueness theorem of the cauchy problem for schrodingers equation in weighted sobolev spaces dan, yuya, differential and integral equations, 2005. On the cauchy problem for the generalized shallow water wave equation article pdf available in journal of differential equations 2457. If a thermal conductivity does not depend on temperature, we have the linear equation. Global solution to the cauchy problem of the nonlinear. The cauchy problem for the hyperbolic semilinear equation. Singbal no part of this book may be reproduced in any form by print, micro. 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. 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. In this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form. Eigenvalues of the laplacian poisson 333 28 problems. Secondorder hyperbolic partial differential equations wave equation linear wave equation. The literature on semilinear wave equations is vast, yet we have complete existence results for only some special cases of semilinearities. The cauchy problem for differential equations a guide to.
The hyperbolic equations constitute a broad class of equations for which the cauchy problem is wellposed. For derivation of the equation and physical meaning of boundary conditions, check salsas book. Every solution of the wave equation utt c2uxx has the form ux, t fx. The mathematics of pdes and the wave equation mathtube. Separation of variables heat equation 309 26 problems. Find the solution of the cauchy problem for the pde. Eigenvalues of the laplacian laplace 323 27 problems. Homogeneous euler cauchy equation can be transformed to linear constant coe cient homogeneous equation by changing the independent variable to t lnx for x0. 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. We now verify that this solution formula indeed yields a solution of the nonhomogeneous wave equation.
The solution to the initial value problem is ux,t e. Cauchy boundary conditions are analogous to the initial conditions for a. 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. Since we assume that the bicharacteristic curve is not locally contained in w f u, there is a point.
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. On the global cauchy problem for the nonlinear schrodinger. Cauchy problem of semilinear inhomogeneous elliptic. 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. Now let us find the general solution of a cauchy euler equation. Cauchy problem of semilinear inhomogeneous elliptic equations of matukumatype with multiple growth terms. It is like having p0x 0 in an ordinary di erential equation. Wave equation cauchy problem cauchy data cylindrical wave progressive wave these keywords were added by machine and not by the authors. The cauchy problem for wave equations with non lipschitz.
In this chapter, we prove that cauchy problem for wave equation is. When c 2 the wave forms are bellshaped curves moving to the right at speed 2. 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. The scheme of generating radial solutions to cauchy problems as limits of exterior. Random data cauchy problem for the wave equation on compact. 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. Cauchy type problem for diffusion wave equation with the riemannliouville partial derivative anatoly a. Cauchy problem au 0 in a neighborhood of t with data u yl\ xj2 and gradw xi, xi. We study the inverse cauchy problem for a time fractional di usion wave equation with distributions in righthand sides. Most of you have seen the derivation of the 1d wave equation from newtons and hookes.
The unique solvability of the problem is established. Solve the following initial value problem for the cauchyeuler equation. This paper concerns withthe global classical solution to the cauchy problem of the nonlinear double dispersive wave equation with strong damping. We shall consider wellposed problems for the wave equation in two and three variables. Solutions to pdes with boundary conditions and initial conditions boundary and initial conditions cauchy, dirichlet, and neumann conditions wellposed problems existence and uniqueness theorems dalemberts solution to the 1d wave equation solution to the. Thus, our initial or cauchy data are ux,0 gx,uhx,x. 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. We prove an existence result for the solution of the cauchy problem and present some. Modeling the longitudinal and torsional vibration of a rod, or of sound waves. 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. 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. Chapter 6 partial di erential equations most di erential equations of physics involve quantities depending on both. While this solution can be derived using fourier series as well, it is. Cauchy problem for a first order quasi linear pde youtube.
In this case the cauchy problem is global in nature, but the condition that be noncharacteristic is no longer sufficient. The selection of permissable wavenumbers k that apply in a particular problem will be determined by solving the appropriate eigenvalue problem. A typical hyperbolic equation is the wave equation. However vx,t 0 is also a solution to the same cauchy problem. 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. 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. By analogy with the cauchy problem for second order o. 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. The cauchy problem for the wave equation using the. Stability of the cauchy problem for wave equations 5. The global cauchy problem for the non linear schrodinger.
In this session we solve cauchy problems for wave equations. 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. 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. Elliptic cauchy problem for one parameter family of domains 4 3. The wave equation in a general spherically symmetric black hole geometry masarik, matthew, advances in theoretical and mathematical physics, 2011. Separation of variables wave equation 305 25 problems. For more complicated examples, specialised software manipulations. A differential equation in this form is known as a cauchy euler equation. The maximum difference between the initial functions is 1n which gets smaller and smaller as n grows. Pdf blow up of solutions to the cauchy problem for.
Cauchy problems for wave equations updated to maple 7. The cauchy problem for the wave equation in a bounded domain let the cauchy problem 1 and 2 be formulated for. Consider the cauchy problem for the quasilinear equation in two variables a. 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. The cauchy problem and wave equations springerlink. Levine and grozdena todorova communicated by david s. 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. Bourgain and bulut 46 studied gibbs measure evolution in radial nonlinear wave on a three dimensional ball. Cauchy data 197 if this expression is zero, we are stuck. The solution to the cauchy problem with ux,01 for all x is ux,t1 for all x and t 0. We have solved the wave equation by using fourier series. Integral surface passing through a given curve 1 cauchy problem for a first order quasi linear pde. If a function f is analytic on a simply connected domain d and c is a simple closed contour lying in d then. Some exact solutions of a heat wave type of a nonlinear.
Jim lambers mat 606 spring semester 201516 lecture 12 and notes these notes correspond to section 4. Burq and tzvetkov 10 obtained the global existence result of the cauchy problem for a supercritical wave equation. Consider the cauchy problem for the wave equation in rn, namely. The cauchy problem for the nonhomogeneous wave equation. Second order homogeneous cauchy euler equations consider the homogeneous differential equation of the form. 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. Consider the cauchy problem for the quasilinear equation in two variables. Random data cauchy problem for the wave equation on. This video shows how to deal with cauchy problem for inhomogeneous second order differential equation with constant coefficients. Global solution to the cauchy problem of the nonlinear double dispersive wave equation with strong damping c.
The cauchy problem for partial differential equations of. The initial value problem cauchy problem for the 1dimenisonal wave equation is given by utt. Burq and tzvetkov 11 established the probabilistic wellposedness for 1. On the cauchy problem for the wave equation on timedependent domains gianni dal maso and rodica toader abstract. The cauchy problem is also natural for some partial differential equations, like the heat and wave equations. Since we assumed k to be constant, it also means that material properties.
Thus a cauchy problem may have more than one solution. We identify a weak spacetime integrability property stip of the solutions and prove that it is. This process is experimental and the keywords may be updated as the learning algorithm improves. In 4 we obtain the longtime behaviour of the solutions of such a cauchy problem, showing that the solutions break. Cauchy problems advanced engineering mathematics 5 7.
But it is often more convenient to use the socalled dalembert solution to the wave equation 1. Pdf on the cauchy problem for the generalized shallow water. Applications other applications of the onedimensional wave equation are. There is given a revision of the formulation and the proof of the theorem regarding the global unique solvability in the class of weak energy solutions of the cauchy problem, for a secondorder semilinear pseudodifferential hyperbolic equation on a smooth riemannian manifold of dimension n. Vilnius university, faculty of mathematics and informatics, naugarduko 24, vilnius, lithuania. 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. 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. Our plan is to identify the real and imaginary parts of f, and then check if the cauchy riemann equations hold for them. In the preceding examples, we note that f x is continuous, but not. Lectures on cauchy problem by sigeru mizohata notes by m. The cauchy problem for wave equations with non lipschitz coefficients ferruccio colombini. As suggested by our terminology, the wave equation 1. Pdf the global cauchy problem for the critical nonlinear. The cauchy problem for elliptic equations 29 references 40 1.
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