On self-similar solutions of semilinear wave equations in higher space dimensions

https://doi.org/10.1016/j.amc.2011.04.039Get rights and content

Abstract

In this paper we analyze self-similar solutions of the semilinear wave equation Φtt  ΔΦ  Φp = 0 for n > 3 space dimensions. We found several classes of analytic solutions labeled by a single parameter, the form of which differ in the vicinity of the light cone. We also propose suitable numerical methods to study them.

Introduction

In many nonlinear PDEs blowup occurs, i.e., singularity is formed in finite time in evolution starting with smooth initial conditions. Blowup is almost always connected with an interesting physical phenomenon (see [1] and references therein) and it is often well described in terms of self-similarity. Spherically symmetric self-similar solutions are defined byΦ(t,r)=(T-t)-αu(ρ),ρ=rT-t,where r = x∣ is the radius. This is a special type of the general self-similar solutions Φλ = λαΦ(t/λ, x/λ). Assuming that the similarity profile u(ρ) is analytic, we obtain blowup when t  T, therefore, it is crucial to have the detailed description of self-similar profiles.

The example of PDE where blowup occurs is the semilinear wave equation with power nonlinearityΦtt-ΔΦ-Φp=0,Φ=Φ(x,t),xRn,which will be examined in this paper. Solutions depend on two parameters p and n, which are integer numbers by assumption. In this paper we limit ourselves to n > 3 and odd p > 2. If p is even, the nonlinear term Φp should be replaced by ∣Φp−1Φ to keep the reflection symmetry for Φ.

The Eq. (2) has the energy functional in the form [7]E[Φ]=RnΦt2+Φ2-1p+1Φp+1dnx,which scales on the general self-similar solutions as E[Φλ] = λβE[Φ], where β=(n-2)p-(n+2)p-1. In general, scaling is called subcritical for β < 0, supercritical for β > 0 and critical when β = 0. In our case the scaling is critical whenp=pQ=n+2n-2.

There is a vast literature that considers various aspects related to the Eq. (2) and its generalizations, e.g., [2], [3], [4], [5], [6], [7], [8]; for scattering method approach see, e.g., [9]. Strichartz estimates and their applications are given in [10], [11]. The application of Besov space method is presented in [12], [13]. We encourage the interested reader to study them. We will use (2) only as a starting point in the search of its self-similar solutions.

It is well known that the Eq. (2) has two particular solutions [14]. The first one is obtained when we neglect coordinate dependence and solve the resulting ODEΦ0(t)=b0(T-t)α,b0=2(p+1)(p-1)21p-1,α=2p-1,T>0with the corresponding constant profileu0(ρ)=b0.The second one is the static spherically symmetric solution of the formΦ(r)=br-α,b=2(p(n-2)-n)(p-1)21p-1and the associated self-similar profileu(ρ)=bρ-αis unbounded when ρ tends to 0. The other self-similar profiles (we will also use the name self-similar solutions for profiles) are obtained from second order ordinary differential equation for similarity profile =ddρ(1-ρ2)u+n-1ρ-2(p+1)p-1ρu-2(p+1)(p-1)2u+up=0,which results from substituting (1) into (2). Casual structure and finite speed of propagation introduced by (2) makes the blowup at the point (t = T, r = 0) only connected with its past light cone, which corresponds to the interval ρ  [0; 1]. This is a main reason to investigate the existence of global analytic self-similar profiles of (9) on this interval. That solutions of (9), if exist, connect two singular points ρ = 0 and ρ = 1 of this equation along analytic curve.

The paper is organized as follows. Starting from (9), we will study its analytic solutions. We begin from local existence theorems at ρ = 0 and ρ = 1 in Section 2. Asymptotics at ρ = 0 is a generalization of the results from [14], but at ρ = 1 we obtain several new classes of solution, which depend on a value of n, p and differ from each other in analytical form. Then in Section 3, generalizing [14], we try to match these two asymptotics to obtain an analytic solution in the entire interval ρ  [0; 1]. This matching is only possible for some special initial data at both endpoints and values of n, p; the requirement of smooth matching is a sort of quantization condition for the values of solution parameters. Solutions with these quantized parameters obey remarkable scaling laws derived in Section 4. Apart form those known in the n = 3 case we observe also qualitatively new shapes of this scaling. We also propose a special numerical method which is best suitable to explain the global existence for one class of new-found solutions. In our studies we will be using mainly asymptotic methods (see, e.g., [15], [16]).

Section snippets

Local existence

In this section we examine local analytic solutions of (9) at both endpoints of [0; 1]. We will be searching for power series solutions which will be easily obtained by employing the Cauchy product [17]l=0al(x-x0)lp=l=0cl(x-x0)l,c0=a0p,cm=1ma0l=1m(lp-m+l)alcm-lfor m > 0, which simplifies nonlinear term in (9).

Global existence

In the previous section we have demonstrated the local existence of two families of solutions of the Eq. (9) – analytic at ρ = 0 and ρ = 1, respectively. One may consider the possibility of matching these two local solutions and, as a result, the existence of the solution which is analytic globally on the whole interval [0; 1]. For n = 3 such solutions were found in [14]. The example of analytical result confirming the existence of global solution is the theorem 1 from [14] which reads.

Theorem 1

[14]

For any odd p  7

Scaling laws

Global analytic solutions on [0; 1] are described by a countable set of pairs of the initial data {(cl,bl)}l=0. From the previous section one can observe that the parameters bl lay on a spiral. Therefore, for large l the consecutive values of bl and its companion cl should obey scaling laws. They were introduced for n = 3 in [14]. In this section we derive a generalization of this laws for noninteger k values, however, when k is integer we obtain a new form of scaling laws. This new form of

Conclusions

We show that a countable set of global analytic solutions on [0; 1] also exist in greater than three space dimensions. Moreover, for some special combinations of space dimensions and nonlinearity parameter we found new classes of solutions, that do not exist in three-dimensional space. These new classes are examples that a second order nonlinear differential equation may have boundary data which specify initial value not only for the function or its firs derivative but also for higher

Acknowledgments

The author thanks Piotr Bizoń and Tadeusz Chmaj for their support during the work. Many parts of this paper are the development of ideas which appeared during inspiring discussions with them. The author thanks also referees for their valuable suggestions that helped to improve this paper. The research was partially carried out with the “Deszno” supercomputer purchased thanks to the financial support of the European Regional Development Fund in the frame-work of the Polish Innovation Economy

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