# Asymptotic profiles of solutions

for structural damped wave equations

###### Abstract

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations , where and . Our result is the approximation formula of the solution by a constant multiple of a special function as , which states that the asymptotic profiles of the solutions are classified into patterns depending on the values and .

Keywords: nonlinear wave equation, fractional damping, the Cauchy problem, critical exponent, asymptotic profile,

2010 Mathematics Subject Classification. Primary 35L15, 35L05; Secondary 35B40

## 1 Introduction

In this paper, we consider the initial value problem for the following equations

(1.1) |

where ,
is a constant,
and are
given initial data.

To begin with, let us introduce several related works to our problem (1.1). In the case when (i.e., strong damping case) one should make mention to some pioneering decay estimates of solutions due to Ponce [16] and Shibata [17], in which Ponce [16] dealt with rather special initial data such as to avoid some singularity, and Shibata [17] has established - decay estimates of solutions. Karch [13] studied an asymptotic self-similar profile of the solution as in the case when , and Ikehata [8] has derived total energy decay estimates of solutions to problem (1.1) with considered in the exterior of a bounded obstacle. While, Lu-Reissig [14] studied the parabolic effect in high order (total) energy estimates to problem (1.1) with damping replaced by , and it seems that recent active researches concerning structural damped waves have their origin in [14], however, in [14] they did not investigate any asymptotic profiles of solutions. Recently, Ikehata-Todorova-Yordanov [12] have discovered its profile of solutions in asymptotic sense as , and it should be mentioned that their result has been established as an abstract theory including (1.1) with , so that it includes quite wide applications. After [12], Ikehata [9] re-studied the problem (1.1) with to observe optimal decay estimates of solutions in terms of -norm. The result of [9] has its motivation in [12], and especially in Ikehata-Natsume [10], in there they studied more precise decay estimates of the total energy and -norm of solutions to the present problem (1.1) by employing the energy method in the Fourier space developed by Umeda-Kawashima-Shizuta [18]. Although the result of [10] has a gap near , soon after [10] the gap has been completely embedded in Charaõ-da Luz-Ikehata [2] by developing a powerful tool to get energy decay estimates.

While, quite recently, in a series of papers due to D’Abbicco [3], D’Abbicco-Ebert [4, 5, 6], D’Abbicco-Reissig [7] and Narazaki-Reissig [15] they have studied several decay estimates and asymptotic profiles of solutions to problem (1.1) in terms of the -norms (), but their main concern seems to be a little restrictive to the case for , i.e., a effective damping case of the problem (1.1) is mainly studied, and so a non-effective damping aspect for the region to problem (1.1) seems to be less investigated at present.

Our main purpose is to classify all asymptotic profiles of solutions to problem (1.1) in terms of the constant and . Especially, our results below essentially seem new in the noneffective damping case for as compared with a previous result due to D’Abbicco-Reissig [7, Theorem 8]. In fact, our results below state about the asymptotic profile of the solution to problem (1.1) in terms of the higher order derivatives, and as a result optimal decay order of the solution can be derived from the viewpoint of the higher order derivatives in -sense.

To state our results, we introduce some notation, which will be used in this paper.

(1.2) |

(1.3) |

We first mention the unique existence of the solution with decay properties to problem (1.1).

###### Proposition 1.1.

Let

(1.4) |

and . Suppose that . Then, there exists a unique solution to problem (1.1) satisfying

(1.5) | |||

(1.6) |

for and , where .

Our next aim is to approximate the solution to (1.1) by a constant multiple of the special functions with a suitable lower bound. We can now formulate our main results.

###### Theorem 1.2.

Under the same assumptions as in Proposition 1.1, it holds that

(1.7) | |||

(1.8) | |||

(1.9) |

as , where

(1.10) |

Moreover there exists such that

(1.11) | |||

(1.12) |

for large , where , and .

If , we can assert the following series of approximation formulas of the solution to (1.1).

###### Theorem 1.3.

Let , , and . If and , then it holds that

(1.13) | |||

(1.14) | |||

(1.15) |

as , where

Moreover, there exists such that

for large , where , and .

###### Remark 1.4.

Before closing this section, we summarize notation, which is used throughout this paper. Let denote the Fourier transform of defined by

with . Also, let or denote the inverse Fourier transform.

For , let be the Sobolev space;

where is the usual Lebesgue space for . For the notation of function spaces, the domain is often abbreviated, and we frequently use the notation without confusion. Furthermore, in the following, denotes a positive constant, which may change from line to line.

This paper is organized as follows. In section 2, we set up notation of the solution formula by the Fourier multiplier expression, which is useful to describe the asymptotic profiles of solutions. Section 3 describes several results of [7] in terms of our notation. Section 4 is devoted to the study of the behaviors of the Fourier multipliers in the Fourier space. In section 5, we prove the upper bound of the norms of the evolution operators, which mean decay properties. Section 6 provides approximation formulas of the evolution operators of (1.1). In section 7, we prove our main results.

## 2 Solution formula

In this section, we formulate the solution of (1.1) by using the Fourier multiplier theory. We remark that our new ingredient here is the case for and with . It is useful to obtain the asymptotic profile of solutions. The results in this section is essentially obtained by D’Abicco-Reissig [7], however, for the reader’s convenience, we repeat the derivation of the evolution operators to (1.1).

We begin with recalling the characteristic roots of (1.1). Applying the Fourier transform to the equation (1.1), we see

(2.1) |

and we have the characteristic equations . Then we see that the characteristic roots are given by

and roughly speaking, for small , their behaviors are given by

and

Thereafter we introduce radial cut-off functions which will be used in the proofs to aligned to the low-, middle- and high-frequency parts. Let , and be

Here we choose satisfying

(2.2) |

### 2.1 The case for .

When , we can write the solution of (2.1) by using constants and such as

The direct calculation implies

where

Therefore we obtain the following Fourier multiplier expression of the solution :

(2.3) |

where

(2.4) |

By using the cut-off functions (), we also have the localized operators defined by

(2.5) |

where we denote

(2.6) |

### 2.2 The case for .

For the case , we can choose constants and such as

where

(2.7) |

and this leads to

Namely, we find

(2.8) |

where

(2.9) |

We also introduce the localized operators of () as follows:

(2.10) |

where is defined by

(2.11) |

We continue, in a similar fashion, to obtain the expression of the solution with corresponding to the value of . Namely, we have

for and

for . For simplicity we introduce the notation

(2.12) |

(2.13) |

and

(2.14) |

for . For the case , as was pointed out in several previous results (see e.g. [3], [7] and [15]), we can obtain

and then we define

(2.15) |

and

(2.16) |

for .

Therefore, we have just arrived at the expression of the solution with for (1.1) by

(2.17) |

###### Remark 2.1.

We note that the choice of defined by (2.2) means that the positive root of does not belong to for .

## 3 Restatement of the results by [7]

Our results here are closely related to those of [7]. In this section, we summarize, without proofs, the precise statements of their results, the point-wise estimates of the fundamental solutions for (1.1) in the Fourier space, and decay estimates of the solution for (1.1) by using our notation and terminology introduced in the previous section. The following lemmas show the behavior of for and in the Fourier space.

###### Lemma 3.1.

Let , , and . Then, there exist and such that