In this paper, we give its sharp convergence rate in the weighted norms for a class of initial data. Second order homogeneous cauchyeuler equations consider the homogeneous differential equation of the form. Asymptotic behavior of solutions for some semilinear heat. Singbal no part of this book may be reproduced in any form by print, micro. The cauchy problem is to determine a solution of the equation. The cauchy problem for a nonhomogeneous heat equation with. Diffusion phenomena of solutions to the cauchy problem for. Analytic solutions of partial differential equations university of leeds. A differential equation in this form is known as a cauchyeuler equation. As an example, let us consider the nonlinear heat equations. The cauchy problem for a nonhomogeneous heat equation with reaction. Derive a fundamental so lution in integral form or make use of the similarity properties of the equation to nd the. Pdf blowup problem for semilinear heat equation with nonlinear.
The main goal is to study the influence of the gradient terms on the blowup profile. We consider the cauchy problem for the semilinear heat equation where ut,x. Existence and asymptotic behavior of boundary blowup solutions for weighted p x laplacian equations with exponential nonlinearities. A note on the life span of semilinear pseudoparabolic. The heat conduction is described by the well known, thoroughly investigated equation. We prove the boundedness of global classical solutions for the semilinear heat equation u t. A cauchy problem for the heat equation springerlink. Cauchy problem of semilinear inhomogeneous elliptic. 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.
In this paper we will consider initial data u 0 which do not belong to l. By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Life spans of solutions of the cauchy problem for a semilinear heat equation j. Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation peter pol a cik school of mathematics, university of minnesota, minneapolis, mn 55455, usa eiji yanagida mathematical institute, tohoku university, sendai 9808578, japan abstract we consider the cauchy problem for a semilinear heat equation. The critical fujita number for a semilinear heat equation in exterior domains with homogeneous neumann boundary values volume issue 3 h. Blowup of a degenerate nonlinear heat equation poon, chicheung, taiwanese journal of mathematics, 2011. The cauchy problem for heat equations with exponential. The cauchy problem for the nonhomogeneous wave equation. The cauchy problem for semilinear parabolic equations in besov spaces article pdf available in houston journal of mathematics 303 january 2004 with 154 reads how we measure reads. The literature on semilinear wave equations is vast, yet we have complete existence results for only some special cases of semilinearities.
Cauchy problem and boundary value problems for the heat equation. The cauchy problem for a semilinear heat equation with. 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. Lectures on cauchy problem by sigeru mizohata notes by m. Definitions of different type of pde linear, quasilinear, semilinear, nonlinear. On a system of nonlinear wave equations mengrong li, mengrong li and longyi tsai, longyi tsai, taiwanese journal of mathematics, 2003. The cauchy problem for a semilinear heat equation with singular initial data. On the contrary, solving 2 for the initial data ut. What is the meaning of solving partial differential equations. Asymptotic behavior of solutions to the semilinear wave equation with timedependent damping nishihara, kenji, tokyo. Homogeneous eulercauchy equation can be transformed to linear constant coe cient homogeneous equation by changing the independent variable to t lnx for x0. We consider the cauchy problem for the quasilinear heat equation. Malekformal solutions of the complex heat equation in higher.
Formal solutions of semilinear heat equations sciencedirect. Cauchy problem for semilinear parabolic equations with initial data. The cauchy problem for the semilinear heat equations is studied in the orlicz space exp l2rn, where any power behavior of interaction. Cauchy problem of semilinear inhomogeneous elliptic equations of matukumatype with multiple growth terms. Since we assumed k to be constant, it also means that material properties. Morrey spaces and classification of global solutions for a. In this note, we consider the cauchy problem for the semilinear heat equation in a homogeneous stratified group. Analytic solutions of partial di erential equations. Solutions of a semilinear cauchy problem jaritaskinen abstract. It is well known that if the initial data u 0 belong to l.
For solutions of the cauchy problem and various boundary value problems, see nonhomogeneous heat equation with x,t. Pdf the cauchy problem for a semilinear heat equation. On the cauchy problem for semilinear elliptic equations nguyen huy tuana, tran thanh binhb, tran quoc vietc, daniel lesnicd adepartment of mathematics and applications, sai gon university, ho chi minh city, viet nam bdepartment of mathematics, university of science, vietnam national university, ho chi minh city, viet nam. Long time asymptotics of subthreshold solutions of a. Pdf on the cauchy problem for semilinear elliptic equations. On the cauchy problem for semilinear elliptic equations. Pdf blowup of solutions for semilinear heat equation with. Upper bound estimates for local in time solutions to the. Then we obtain a threshold result for the global existence and nonexistence of solutions. 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. Chapter 7 heat equation home department of mathematics. Blowup rate estimates for a system of reactiondiffusion.
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. This is done under three possible initial energy levels, except the nls as it does not have comparison principle. Namely, under some conditions on this system, we consider the upper blowup rate estimates for its blowup solutions and for the gradients. Essentially the same estimates hold for this problem as for the heat equation. This paper is concerned with the blowup properties of cauchy and dirichlet problems of a coupled system of reactiondiffusion equations with gradient terms. Life span of positive solutions for the cauchy problem for. Pdf cauchy problems of semilinear pseudoparabolic equations. In this paper we show that the cauchy problem for the onedimensional heat equation, though nonwell posed in the sense of hadamard, can be solved numerically. We study the behavior of solutions to the cauchy problem for a semilinear heat equation with supercritical nonlinearity.
C 0 r, there exist a maximal existence time t max 0 and a unique solution. The cauchy problem for a semilinear heat equation with singular initial data bernhard ruf and elide terraneo article pdf available july 2004 with 100 reads how we measure reads. We first recall the result on the cauchy problem for a semilinear heat equation. Finally we discuss the asymptotic behavior of the solution. Solving pdes analytically is generally based on finding a change of variable to transform. This is proved without any radial symmetry or sign assumptions, unlike in all the previously known results for the cauchy problem, and under spatial decay assumptions on the initial data that. The critical fujita number for a semilinear heat equation. R, p 3, and nonnegative cauchy data behaves for large t like the solution of the corresponding linear problem plus a small correction of order t. On the cauchy problem for the onedimensional heat equation. 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. The dye will move from higher concentration to lower. The sharp estimate of the lifespan for semilinear wave equation with timedependent damping ikeda, masahiro and inui, takahisa, differential and integral equations, 2019. Haraux, an optimal estimate for the time singular limit of an abstract wave equation, funkcial. Now let us find the general solution of a cauchyeuler equation.
This hypersurface is known as the carrier of the initial conditions or the initial surface. In this paper, we study the cauchy problem of semilinear heat equations. Global existence, nonexistence and asymptotic behavior of. Convergence of anisotropically decaying solutions of a. The blowup rate of solutions of semilinear heat equations u core. We study the cauchy problem for the semilinear parabolic equa tion.
Blowup problem for semilinear heat equation with nonlinear nonlocal neumann boundary condition. It is known that two solutions approach each other if these initial data are close enough near the spatial infinity. Pdf in this paper, we consider a semilinear heat equation. Fujitaon the blowing up of solutions of the cauchy problem for u t. H is the sublaplacian on we prove the nonexistence of global in time solutions for exponents in the subfujita case, that is for 1 cauchy problem instead of cauchy type problem, for the sake of brevity. In pioneer work 1, fujita showed that the exponent plays the crucial role for the existence and nonexistence of the solutions of 1.
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