Home Global existence for time-dependent damped wave equations with nonlinear memory
Article Open Access

Global existence for time-dependent damped wave equations with nonlinear memory

  • Mokhtar Kirane , Ahmad Z. Fino , Ahmed Alsaedi EMAIL logo and Bashir Ahmad
Published/Copyright: November 18, 2023
Download Article (PDF)
Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

In this article, we consider the Cauchy problem for a semi-linear wave equation with time-dependent damping and memory nonlinearity. Conditions for global existence are presented in the energy space H 1 ( R n ) × L 2 ( R n ) , n ≥ 1 .

Keywords: nonlinear damped wave equation; global existence
MSC 2010: 35L71; 35A01

1 Introduction

This article concerns the Cauchy problem for the following semi-linear damped wave equation:

(P) u t t − Δ u + b ( t ) u t = ∫ 0 t ( t − s ) − γ ∣ u ( s ) ∣ p d s , t > 0 , x ∈ R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n ,

where the unknown function u is real-valued, n ≥ 1 , 0 < γ < 1 , and p > 1 . Here, the kernel is weakly singular. The coefficient of the damping term is given as follows:

b ( t ) ≔ b 0 ( 1 + t ) − β ,

with b 0 > 0 , and 0 ≤ β < 1 . Throughout this article, we assume that the initial data are in the energy space

(IC) ( u 0 , u 1 ) ∈ A m ≔ ( L m ( R n ) ∩ H 1 ( R n ) ) × ( L m ( R n ) ∩ L 2 ( R n ) ) ,

where 1 ≤ m ≤ 2 . Here, ‖ ⋅ ‖ q and ‖ ⋅ ‖ H 1   ( 1 ≤ q ≤ ∞ ) stand for the usual L q ( R n ) -norm and H 1 ( R n ) -norm, respectively.

The nonlinear nonlocal term can be considered as an approximation (with suitable change of variables) of the nonlinearity of the following semi-linear damped wave equation:

u t t − Δ u + b ( t ) u t = ∣ u ( t ) ∣ p ,

since the limit

lim γ → 1 s + − γ = Γ ( 1 − γ ) δ ( s )

exists in the distributional sense, where Γ is the Euler gamma function.

Before presenting our main results, let us dwell on the available literature on the topic. Li and Zhou [10] showed in 1995 that the local solution of the equation must blow up in a finite time if n ≥ 2 , 1 < p ≤ 1 + 2 ⁄ n , and the average of the data is positive:

(P1) u t t − Δ u + u t = ∣ u ( t ) ∣ p .

In 2001, Todorova and Yordanov [14] developed a weighted energy method and determined the critical exponent of ( P 1 ) as follows:

p c = 1 + 2 n ,

which is well known as Fujita’s critical exponent for the heat equation u t − Δ u = u p (see [5]). More precisely, they proved small data global existence when p > 1 + 2 ⁄ n and blow-up of all solutions of ( P 1 ) with positive average data when 1 < p < 1 + 2 ⁄ n . Later on, Zhang [16] and Kirane and Qafsasoui [9] independently showed that the critical exponent p = 1 + 2 ⁄ n belongs to the blow-up region. Todorova and Yordanov [14] assumed the data to have compact support and essentially used this property in their analysis. However, in 2005, Ikehata and Tanizawa [7] removed this assumption.

Recently, Nishihara [12] in 2011 and Lin et al. [11] in 2012 considered the semi-linear wave equation with time-dependent damping

u t t − Δ u + b ( t ) u t = ∣ u ∣ p ,

where

b ( t ) = b 0 ( 1 + t ) − β , β ∈ ( − 1 , 1 ) ,

and found the critical exponent

p c = 1 + 2 n .

This shows that time-dependent coefficients of damping term of the above form do not influence the critical exponent. For more results about damped wave equations with memory terms, we refer the interested readers to Fino [4] and D’Abbicco [1,2].

On the other hand, problem ( P ) has recently been studied by Kaddour and Reissig [8]. They proved the small data global (in time) well-posedness of a solution in C ( [ 0 , ∞ ) ; H σ ( R n ) ) when σ ∈ ( 0 , 1 ) and in C ( [ 0 , ∞ ) ; H σ ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) ; H σ − 1 ( R n ) ) for σ ∈ ( 1 , n 2 ) by using the standard energy method and the fractional Gagliardo-Nirenberg inequality. Our goal is to complete this study for σ = 1 by proving the small data global well-posedness of a solution in the standard energy space C ( [ 0 , ∞ ) ; H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) ; L 2 ( R n ) ) by employing the same method. For β = 1 , this result improves the global existence result proved by Fino [4] in 2012. For the problem ( P ) , it has been noted that the standard energy method works better than the weighted energy method developed in the study by Ikehata and Tanizawa [7] and Todorova and Yordanov [14], and used in the study by Lin et al. [12] and and Nishihara [11].

This article is organized as follows: in Section 2, we present some definitions and properties concerning the linear homogeneous and inhomogeneous equations related to ( P ) ; Section 3 is devoted to our main results; and finally, we prove the theorems on global existence (Theorems 3, 4, and 5) in Section 4. Here, we emphasize that we can easily prove Theorems 6–9 in a similar manner. Throughout this article, C denotes a universal positive constant that may change from line to line.

2 Preliminaries

In this section, we give some preliminary properties that will be used in the proof of main results.

In order to define the mild solution of equation (P), we consider the corresponding linear homogeneous equation

(1) u t t − Δ u + b ( t ) u t = 0 , t > 0 , x ∈ R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n .

It is well known that there exists a unique strong solution u of equation (1) for any ( u 0 , u 1 ) ∈ H 2 ( R n ) × H 1 ( R n ) , for instance, see [6, Theorem 2.27]. Let us denote by R ( t , s ) the operator, which maps the initial data ( u ( s ) , u t ( s ) ) ∈ H 2 ( R n ) × H 1 ( R n ) given at time s ≥ 0 to the solution u ( t ) ∈ H 2 ( R n ) at the time t ≥ s , i.e., the solution u of equation (1) is defined by u ( t ) = R ( t , 0 ) ( u 0 , u 1 ) . The operator R ( t , s ) can uniquely be extended as R ( t , s ) : H 1 ( R n ) × L 2 ( R n ) ⟶ C ( [ s , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ s , ∞ ) , H 1 ( R n ) ) (see [15, Appendix]).

Now, we consider the linear inhomogeneous equation

(2) u t t − Δ u + b ( t ) u t = F ( t , x ) , t > 0 , x ∈ R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n .

Definition 1

(Mild solution) Let ( u 0 , u 1 ) ∈ H 1 ( R n ) × L 2 ( R n ) and F ∈ C ( [ 0 , ∞ ) ; L 2 ( R n ) ) . We say that u is a mild solution of equation (2) if u ∈ C ( [ 0 , ∞ ) ; H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) ; L 2 ( R n ) ) and u satisfies the initial data u ( 0 ) = u 0 and u t ( 0 ) = u 1 and the integral equation

u ( t , x ) = R ( t , 0 ) ( u 0 , u 1 ) + ∫ 0 t S ( t , s ) F ( s , x ) d s

in the sense of H 1 , where S ( t , s ) g ≔ R ( t , s ) ( 0 , g ) for g ∈ H 1 ( R n ) .

Proposition 1

[15, Proposition 9.15] Let ( u 0 , u 1 ) ∈ H 1 ( R n ) × L 2 ( R n ) and F ∈ C ( [ 0 , ∞ ) ; L 2 ( R n ) ) , then there exists a unique mild solution u of equation (2).

Nonlinear equations do not always admit global-in-time solutions. Therefore, we consider the solution defined on an interval [ 0 , T ) for T > 0 . We call such a solution as local-in-time solution (or local solution); if T = ∞ , then we call it global-in-time solution (or global solution).

Definition 2

(Mild solution) Let ( u 0 , u 1 ) ∈ H 1 ( R n ) × L 2 ( R n ) . We say that u is a mild solution of ( P ) if

u ∈ C ( [ 0 , T ) ; H 1 ( R n ) ) ∩ C 1 ( [ 0 , T ) ; L 2 ( R n ) )

satisfies the initial data u ( 0 ) = u 0 and u t ( 0 ) = u 1 and the integral equation

(3) u ( t , x ) = R ( t , 0 ) ( u 0 , u 1 ) + Γ ( δ ) ∫ 0 t S ( t , τ ) J 0 ∣ τ δ ( ∣ u ∣ p ) d τ ≕ u lin ( t ) + u n l ( t )

in the sense of H 1 , where J 0 ∣ τ δ is the Riemann-Liouville left-sided fractional integral (e.g., see [13, Chapter 1]) of order δ = 1 − γ ∈ ( 0 , 1 ) defined by

J 0 ∣ τ δ ( ∣ u ∣ p ) = 1 Γ ( δ ) ∫ 0 τ ( τ − s ) − γ ∣ u ( s ) ∣ p d s .

Set

a = a m ≔ n 2 1 m − 1 2

and

I 0 = ‖ ( u 0 , u 1 ) ‖ A m = ‖ u 0 ‖ m + ‖ u 0 ‖ H 1 + ‖ u 1 ‖ m + ‖ u 1 ‖ 2 .

The following theorems present key estimate on u lin and u n l defined in equation (3).

Theorem 1

([3, Theorem A])

‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) ,

‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( a + 1 2 ) ( β + 1 ) ,

and

‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) − 1 .

Set

B ( t , τ ) = ∫ τ t 1 b ( τ ) d τ = B ( t , 0 ) − B ( τ , 0 ) ,

and

B ( t , 0 ) = ∫ 0 t 1 b ( τ ) d τ = ( 1 + t ) β + 1 − 1 .

Theorem 2

([3, Theorem 3])

‖ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ‖ J 0 ∣ τ δ ( ∣ u ∣ p ) ‖ L m ∩ L 2 d τ ,

‖ ∇ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ‖ J 0 ∣ τ δ ( ∣ u ∣ p ) ‖ L m ∩ L 2 d τ

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ‖ J 0 ∣ τ δ ( ∣ u ∣ p ) ‖ L m ∩ L 2 d τ .

The following lemma plays a crucial role in the proof of global existence theorems.

Lemma 1

[8, Lemma 5.1] Assume that 0 < γ < 1 , a ≥ 0 , b > 1 , and r ∈ ( − 1 , 1 ) . Then, we have

∫ 0 t ( 1 + τ ) − r ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − b d s d τ ≤ C ( 1 + t ) − min { a ( 1 − r ) ; γ } , if max { a ; γ + r } > 1 , ( 1 + t ) − min { a ( 1 − r ) ; γ } ln ( 2 + t ) , if max { a ; γ + r } = 1 , ( 1 + t ) 1 − a ( 1 − r ) − γ − r , if max { a ; γ + r } < 1 .

3 Main results

For m ≤ 2 , note that

a = n 2 1 m − 1 2 ≥ 0 .

Later, we need the following condition:

2 m ≤ p ≤ n ( n − 2 ) + ,

i.e., we need to suppose for n ≥ 3 that

n ≤ 4 ( 2 − m ) + .

So, by this condition, we have

n ≤ 4 ( 2 − m ) + ≤ 4 m ( 2 − m ) + ,

which implies that

0 ≤ a = n 2 1 m − 1 2 ≤ 1 .

Due to technical reasons, we study three cases: γ − β < 1 , γ − β = 1 , and γ − β > 1 .

We start with the case: γ − β < 1 , where we have to distinguish the following cases: a ∈ [ 1 ⁄ 2 , 1 ) , a ∈ [ 0 , 1 ⁄ 2 ) , and a = 1 .

Theorem 3

(Global existence: case of γ − β < 1 and a ∈ [ 1 ⁄ 2 , 1 ) ) Let n ≥ 2 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ) , 1 2 ≤ a < 1 , and γ − β < 1 . Suppose that n ≤ 4 2 − m when n ≥ 3 , and

2 m ≤ p ≤ n ( n − 2 ) + .

If

p > 2 + 2 n m ( 1 − a ) ( β + 1 ) ( n − 2 ) ( 1 − a ) ( β + 1 ) + 2 γ ≕ p 1 * ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ,

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ , if a > 1 2 , ( 1 + t ) − γ ln ( 2 + t ) , if a = 1 2 ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ .

Remark

We exclude the case n = 1 in Theorem 3 because

a = 1 2 1 m − 1 2 ≤ 1 4 .

Theorem 4

(Global existence: case of γ − β < 1 and a ∈ [ 0 , 1 ⁄ 2 ) ) Let n ≥ 1 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ] , a < 1 2 , and γ − β < 1 . Suppose that n ≤ 4 ( 2 − m ) + when n ≥ 3 , and

2 m ≤ p ≤ n ( n − 2 ) + .

We also suppose that γ > ( 2 m − n ) ( β + 1 ) ⁄ 2 m . If

p > 2 m + n ( β + 1 ) ( n − 2 m ) ( β + 1 ) + 2 m γ ≕ p 2 * ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ,

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 2 − a ) ( β + 1 ) − γ ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ , if a > 0 , ( 1 + t ) β − γ ln ( 2 + t ) , if a = 0 .

Theorem 5

(Global existence: case of γ − β < 1 and a = 1 ) Let n = 4 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ) , a = 1 , and γ − β < 1 . Suppose that

2 m ≤ p ≤ n n − 2 .

If

p > 1 γ ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ,

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ .

Next, we consider the case when γ − β > 1 .

Theorem 6

(Global existence: case of γ − β > 1 ) Let n ≥ 1 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ) , 0 ≤ a ≤ 1 , and γ − β > 1 . Suppose that n ≤ 4 2 − m when n ≥ 3 , and

2 m ≤ p ≤ n ( n − 2 ) + .

If

p > 2 + 2 n m ( 1 − a ) ( β + 1 ) ( n − 2 ) ( 1 − a ) ( β + 1 ) + 2 γ = p 1 * ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) − a ( β + 1 ) ,

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − min { ( a + 1 2 ) ( β + 1 ) ; γ } ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) − min { ( a + 1 ) ( β + 1 ) ; γ } − β .

Finally, for the case of γ − β = 1 , we have to distinguish three cases: a = 0 , a ∈ ( 0 , 1 ⁄ 2 ] , and a ∈ ( 1 ⁄ 2 , 1 ] .

Theorem 7

(Global existence: case of γ − β = 1 and a = 0 ) Let n ≥ 2 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , a = 0 (i.e., m = 2 ), and γ − β = 1 . Suppose that

1 < p ≤ n ( n − 2 ) + .

If

p > 2 + 2 n m ( 1 − a ) ( β + 1 ) ( n − 2 ) ( 1 − a ) ( β + 1 ) + 2 γ = p 1 * ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ln ( 2 + t )

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ 2 ln ( 2 + t ) ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) − 1 ln ( 2 + t ) .

Theorem 8

(Global existence: case of γ − β = 1 and a ∈ ( 0 , 1 ⁄ 2 ] ) Let n ≥ 1 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ] , a ∈ ( 0 , 1 ⁄ 2 ] , and γ − β = 1 . Suppose that n ≤ 4 ( 2 − m ) + when n ≥ 3 , and

2 m ≤ p ≤ n ( n − 2 ) + .

We also suppose that γ > ( 2 m − n ) ( β + 1 ) ⁄ 2 m . If

p > 2 m + n ( β + 1 ) ( n − 2 m ) ( β + 1 ) + 2 m γ = p 2 * ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) − a γ ln ( 2 + t )

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − ( a + 1 2 ) γ ln ( 2 + t ) ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) − 1 .

Theorem 9

(Global existence: case of γ − β = 1 and a ∈ ( 1 ⁄ 2 , 1 ] ) Let n = 4 , γ ∈ ( 0 , 1 ) , 0 ≤ β < 1 , m ∈ [ 1 , 2 ) , a ∈ ( 1 ⁄ 2 , 1 ] , and γ − β = 1 . Suppose that

2 m ≤ p ≤ n n − 2 .

If

p > 1 γ ,

then there exists a small enough positive constant 0 < ε 0 ≪ 1 such that, for any initial data satisfying

I 0 ≤ ε 0 ,

there is a uniquely global (in time) mild solution

u ∈ C ( [ 0 , ∞ ) , H 1 ( R n ) ) ∩ C 1 ( [ 0 , ∞ ) , L 2 ( R n ) ) .

Moreover, the solution satisfies the following estimates:

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) − a γ ln ( 2 + t )

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ,

and

‖ ∂ t u ( t ) ‖ 2 ≤ C ( 1 + t ) − 1 .

4 Proof of Theorems 3–5

Proof of Theorem 3

We start by introducing, for T > 0 , the space of energy solutions

X ( T ) = C ( [ 0 , T ] , H 1 ( R n ) ) ∩ C 1 ( [ 0 , T ] , L 2 ( R n ) )

equipped with the norm

‖ v ‖ X ( T ) = sup 0 ≤ t ≤ T { ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ v ( t ) ‖ 2 + ( 1 + t ) γ ‖ ∇ v ( t ) ‖ 2 + ( 1 + t ) γ − β ‖ ∂ t v ( t ) ‖ 2 }

when a > 1 2 , and

‖ v ‖ X ( T ) = sup 0 ≤ t ≤ T { ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ v ( t ) ‖ 2 + ( 1 + t ) γ ( ln ( 2 + t ) ) − 1 ‖ ∇ v ( t ) ‖ 2 + ( 1 + t ) γ − β ‖ ∂ t v ( t ) ‖ 2 }

when a = 1 2 , for any v ∈ X ( T ) . We will use the Banach fixed-point theorem. Let us define the complete metric space: B R ( T ) = { v ∈ X ( T ) ; ‖ v ‖ X ( T ) ≤ 2 R } , where R > 0 is a positive constant that will be chosen later. We also suppose that I 0 ≤ ε 0 , where 0 < ε 0 ≪ 1 is a positive constant, small enough to be determined later. By Proposition 1, we define a mapping Φ : B R ( T ) → X ( T ) such that

Φ ( u ) ( t ) ≔ Φ ( u ) lin ( t ) + Φ ( u ) n l ( t ) = R ( t , 0 ) ( u 0 , u 1 ) + Γ ( δ ) ∫ 0 t S ( t , τ ) J 0 ∣ τ δ ( ∣ u ∣ p ) d τ for  u ∈ B R ( T ) .

Our goal is to prove that Φ : B R ( T ) ⟶ B R ( T ) is a contraction map.

•   Φ : B R ( T ) ⟶ B R ( T ) . Let u ∈ B R ( T ) .

Case of a > 1 2 .

Estimation of ‖ Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorem 1, and γ − β < 1 , we have

(4) ‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ .

On the other hand, by Theorem 2, we have

‖ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ‖ J 0 ∣ τ δ ( ∣ u ∣ p ) ‖ L m ∩ L 2 d τ ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

The condition

2 m ≤ p ≤ n ( n − 2 ) + ,

allows us to use the following Gagliardo-Nirenberg inequality:

‖ u ( s ) ‖ j p p ≤ C ‖ u ( s ) ‖ 2 p ( 1 − θ ( j p ) ) ‖ ∇ u ( s ) ‖ 2 p θ ( j p ) ≤ C ( 1 + s ) − β j ‖ u ‖ X ( T ) p , j = 2 , m ,

where

θ ( j p ) ≔ n 1 2 − 1 j p and β j ≔ p ( a − 1 ) ( β + 1 ) 1 − n 2 + γ + n j ( a − 1 ) ( β + 1 ) , j = 2 , m .

As m < 2 , we can easily see that

β 2 > β m ,

and then

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ‖ u ‖ X ( T ) p ,

therefore

‖ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ .

Note that β m > 1 ⇔ p > p 1 * , which implies, using the fact that max { a , γ − β } < 1 and Lemma 1, that

(5) ‖ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

By equations (4) and (5), we obtain

(6) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and the fact that a > 1 ⁄ 2 and γ − β < 1 , we obtain

(7) ‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( a + 1 2 ) ( β + 1 ) ≤ C ε 0 ( 1 + t ) − γ

and

‖ ∇ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

Similar to the estimation of ‖ Φ ( u ) ( t ) ‖ 2 , we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ‖ u ‖ X ( T ) p ,

which implies that

‖ ∇ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ .

Note that β m > 1 ⇔ p > p 1 * , which implies, using the fact that max { a + 1 2 , γ − β } = a + 1 2 > 1 and Lemma 1, that

(8) ‖ ∇ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ‖ u ‖ X ( T ) p .

By equations (7) and (8), we obtain

(9) ( 1 + t ) γ ‖ ∇ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ̲ : By Theorems 1 and 2, and the fact that a > 0 and γ − β < 1 , we obtain

(10) ‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) − 1 ≤ C ε 0 ( 1 + t ) β − γ

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

In addition, as above, we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ‖ u ‖ X ( T ) p ,

therefore

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ .

As p > p 1 * ⇔ β m > 1 and using the fact that max { a + 1 , γ − β } = a + 1 > 1 and Lemma 1, we have

(11) ‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ ‖ u ‖ X ( T ) p .

By equations (10) and (11), we obtain

(12) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Summing up equations (6), (9), and (12) and using u ∈ B R ( T ) , we conclude that

‖ Φ ( u ) ‖ X ( T ) ≤ C ε 0 + C ‖ u ‖ X ( T ) p ≤ C ε 0 + C R p .

At this stage, we first choose R > 0 such that C R p − 1 ≤ 1 , and then, we take 0 < ε 0 ≪ 1 such that R ≥ C ε 0 . In consequence, we obtain

‖ Φ ( u ) ‖ X ( T ) ≤ 2 R ,

i.e., Φ ( u ) ∈ B R ( T ) .

Case of a = 1 2 .

Estimation of ‖ Φ ( u ) ( t ) ‖ 2 ̲ : As in the case of a > 1 ⁄ 2 , we have

(13) ‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ

and

‖ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ ,

where

‖ u ( s ) ‖ j p p ≤ C ‖ u ( s ) ‖ 2 p ( 1 − θ ( j p ) ) ‖ ∇ u ( s ) ‖ 2 p θ ( j p ) , j = 2 , m .

Due to the condition a = 1 2 , we obtain

‖ ∇ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ‖ u ‖ X ( T ) ≤ C ( 1 + t ) − γ + δ ‖ u ‖ X ( T )

for δ ≪ 1 to be chosen later. Therefore, we have

‖ u ( s ) ‖ j p p ≤ C ( 1 + s ) − β j + δ p θ ( j p ) ‖ u ‖ X ( T ) p , j = 2 , m .

As m < 2 , we can easily see that β 2 > β m and θ ( m p ) < θ ( 2 p ) , and so

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m + δ p θ ( 2 p ) ‖ u ‖ X ( T ) p .

Thus, we obtain

‖ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ .

Note that β m > 1 ⇔ p > p 1 * , so by choosing δ ≪ 1 , namely δ < ( β m − 1 ) ⁄ ( p θ ( 2 p ) ) , we obtain β m − δ p θ ( 2 p ) > 1 , which implies, using the fact that max { a , γ − β } < 1 and Lemma 1, that

(14) ‖ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

By equations (13) and (14), we arrive at

(15) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) ‖ 2 ̲ : By Theorems 1 and 2, and the fact that a = 1 ⁄ 2 and γ − β < 1 , we obtain

(16) ‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( β + 1 ) ≤ C ε 0 ( 1 + t ) − γ ln ( 2 + t )

and

‖ ∇ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

Similar to the estimation of ‖ Φ ( u ) ( t ) ‖ 2 , we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m + δ p θ ( 2 p ) ‖ u ‖ X ( T ) p ,

therefore

‖ ∇ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ .

As above, by choosing δ ≪ 1 , namely δ < ( β m − 1 ) ⁄ ( p θ ( 2 p ) ) , we obtain β m − δ p θ ( 2 p ) > 1 , which implies, using the fact that max { 1 , γ − β } = 1 > γ and Lemma 1, that

(17) ‖ ∇ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ‖ u ‖ X ( T ) p .

By equations (16) and (17), we obtain

(18) ( 1 + t ) γ ( ln ( 2 + t ) ) − 1 ‖ ∇ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and the fact that a > 0 , β ≥ 0 , we obtain

(19) ‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) − 1 ≤ C ε 0 ( 1 + t ) β − γ

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

In the same way, as before, we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m + δ p θ ( 2 p ) ‖ u ‖ X ( T ) p ,

therefore

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ .

As above, using the same choice of δ and the fact that β m > 1 , we have β m − δ p θ ( 2 p ) > 1 , which implies, using max { a + 1 , γ − β } = a + 1 > 1 and Lemma 1, that

(20) ‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ ‖ u ‖ X ( T ) p .

By equations (19) and (20), we obtain

(21) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Summing up equations (15), (18), and (21) and using the fact that u ∈ B R ( T ) , we conclude that

‖ Φ ( u ) ‖ X ( T ) ≤ C ε 0 + C ‖ u ‖ X ( T ) p ≤ C ε 0 + C R p .

By choosing R > 0 such that C R p − 1 ≤ 1 , and 0 < ε 0 ≪ 1 such that R ≥ C ε 0 , we arrive at

‖ Φ ( u ) ‖ X ( T ) ≤ 2 R ,

i.e., Φ ( u ) ∈ B R ( T ) .

•   Φ is a contraction. Let u , v ∈ B R ( T ) .

Case of a > 1 2 .

Estimation of ‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ̲ : By Theorem 2, we have

(22) ‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ⋅ ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

Using the estimation

∣ ∣ u ∣ p − ∣ v ∣ p ∣ ≤ C ( p ) ∣ u − v ∣ ( ∣ u ∣ p − 1 + ∣ v ∣ p − 1 )

and Hölder’s inequality, we obtain

‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ j ≤ ‖ u ( s ) − v ( s ) ‖ j p ( ‖ u ( s ) ‖ j p p − 1 + ‖ v ( s ) ‖ j p p − 1 ) , j = 2 , m .

Let us estimate ‖ u ( s ) − v ( s ) ‖ j p for j = 2 , m . By Gagliardo-Nirenberg inequality, we obtain

‖ u ( s ) − v ( s ) ‖ j p ≤ C ‖ u ( s ) − v ( s ) ‖ 2 1 − θ ( j p ) ‖ ∇ ( u ( s ) − v ( s ) ) ‖ 2 θ ( j p ) ≤ C ( 1 + s ) ( 1 − θ ( j p ) ) ( 1 − a ) ( β + 1 ) − γ ‖ u − v ‖ X ( T ) .

Similarly, we have

‖ u ( s ) ‖ j p p − 1 ≤ C ‖ u ( s ) ‖ 2 ( 1 − θ ( j p ) ) ( p − 1 ) ‖ ∇ u ( s ) ‖ 2 θ ( j p ) ( p − 1 ) ≤ C ( 1 + s ) ( p − 1 ) ( ( 1 − θ ( j p ) ) ( 1 − a ) ( β + 1 ) − γ ) ‖ u ‖ X ( T ) p − 1

and

‖ v ( s ) ‖ j p p − 1 ≤ C ‖ v ( s ) ‖ 2 ( 1 − θ ( j p ) ) ( p − 1 ) ‖ ∇ v ( s ) ‖ 2 θ ( j p ) ( p − 1 ) ≤ C ( 1 + s ) ( p − 1 ) ( ( 1 − θ ( j p ) ) ( 1 − a ) ( β + 1 ) − γ ) ‖ v ‖ X ( T ) p − 1 .

Therefore, we conclude that

(23) ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ j ≤ C ( 1 + s ) − β j ( ‖ u ‖ X ( T ) p − 1 + ‖ v ‖ X ( T ) p − 1 ) ‖ u − v ‖ X ( T ) , j = 2 , m .

Plugging equation (23) into equation (22), we arrive at

‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ≤ C ( ‖ u ‖ X ( T ) p − 1 + ‖ v ‖ X ( T ) p − 1 ) ‖ u − v ‖ X ( T ) ⋅ ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a × ∫ 0 τ ( τ − s ) − γ ( ( 1 + s ) − β m + ( 1 + s ) − β 2 ) d s d τ .

As m < 2 , we can easily see that β 2 > β m , and then

‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ ,

where we have used the fact that u , v ∈ B R ( T ) . Note that β m > 1 ⇔ p > p 1 * , which implies, using the fact that max { a , γ − β } < 1 and Lemma 1, that

‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ R p − 1 ‖ u − v ‖ X ( T ) .

Hence, we have

(24) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ̲ : By Theorem 2, we have

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

As in the estimation of ‖ ( Φ ( u ) − Φ ( v ) ) ( t ) ‖ 2 , we obtain

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ .

Moreover, as β m > 1 ⇔ p > p 1 * , it follows by max { a + 1 2 , γ − β } = a + 1 2 > 1 and Lemma 1 that

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C ( 1 + t ) − γ R p − 1 ‖ u − v ‖ X ( T ) ,

i.e.,

(25) ( 1 + t ) γ ‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Estimation of ‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ̲ : By Theorem 2, we have

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ⋅ ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

Again, using the foregoing arguments, we obtain

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m d s d τ .

In addition, β m > 1 ⇔ p > p 1 * , which implies, using the fact that max { a + 1 , γ − β } = a + 1 > 1 and Lemma 1, that

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ( 1 + t ) β − γ ‖ u − v ‖ X ( T ) ,

i.e.,

(26) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Summing up equations (24), (25), and (26), we conclude that

‖ Φ ( u ) − Φ ( v ) ‖ X ( T ) ≤ C R p − 1 ‖ u − v ‖ X ( T ) ,

and therefore

‖ Φ ( u ) − Φ ( v ) ‖ X ( T ) ≤ 1 2 ‖ u − v ‖ X ( T ) ,

where R > 0 has been chosen such that C R p − 1 ≤ 1 2 . This establishes that Φ is a contraction.

Case of a = 1 2 .

Estimation of ‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ̲ : By Theorem 2, we obtain

‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

As a = 1 2 , we have

‖ ∇ u ( t ) − ∇ v ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ‖ u − v ‖ X ( T ) ≤ C ( 1 + t ) − γ + δ ‖ u − v ‖ X ( T ) ,

where δ ≪ 1 has been chosen as above, namely δ < ( β m − 1 ) ⁄ ( p θ ( 2 p ) ) . Therefore, as in the case of a > 1 2 , we obtain

‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a × ∫ 0 τ ( τ − s ) − γ ( ( 1 + s ) − β m + δ p θ ( m p ) + ( 1 + s ) − β 2 + δ p θ ( 2 p ) ) d s d τ .

As m < 2 , we can easily see that β 2 > β m and θ ( m p ) < θ ( 2 p ) , and therefore,

‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ .

Note that β m > 1 ⇔ p > p 1 * , so by our choice of δ , we obtain β m − δ p θ ( 2 p ) > 1 , which implies, using the fact that max { a , γ − β } < 1 and Lemma 1, that

‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ‖ u − v ‖ X ( T ) ,

i.e.,

(27) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ Φ ( u ) ( t ) − Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Estimation of ‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ̲ : By Theorem 2, we have

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

Likewise, we arrive at

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ ,

which implies, using the fact that β m − δ p θ ( 2 p ) > 1 , max { 1 , γ − β } = 1 > γ , and Lemma 1, that

‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ( 1 + t ) − γ ln ( 2 + t ) ‖ u − v ‖ X ( T ) ,

i.e.,

(28) ( 1 + t ) γ ( ln ( 2 + t ) ) − 1 ‖ ∇ Φ ( u ) ( t ) − ∇ Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Estimation of ‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ̲ : By Theorem 2, we obtain

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 × ∫ 0 τ ( τ − s ) − γ ( ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ m + ‖ ∣ u ( s ) ∣ p − ∣ v ( s ) ∣ p ‖ 2 ) d s d τ .

By using the same arguments as before, we conclude that

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m + δ p θ ( 2 p ) d s d τ ,

which implies, using β m − δ p θ ( 2 p ) > 1 , that

‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ( 1 + t ) β − γ ‖ u − v ‖ X ( T ) ,

i.e.,

(29) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) − ∂ t Φ ( v ) ( t ) ‖ 2 ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Summing up equations (27), (28), and (29), we conclude that

‖ Φ ( u ) − Φ ( v ) ‖ X ( T ) ≤ C R p − 1 ‖ u − v ‖ X ( T ) .

Choosing R > 0 such that C R p − 1 ≤ 1 ⁄ 2 , we arrive at

‖ Φ ( u ) − Φ ( v ) ‖ X ( T ) ≤ 1 2 ‖ u − v ‖ X ( T ) ,

i.e., Φ is a contraction.

Hence, by the Banach fixed point theorem, there exists a unique mild solution u ∈ X ( T ) to problem ( P ) . This completes the proof of Theorem 3.□

Proof of Theorem 4

The proof of Theorem 4 is very similar to that of Theorem 3. So, the repeated arguments will be omitted. Let us consider, for T > 0 , the same space of energy solutions X ( T ) equipped with the norm

‖ u ‖ X ( T ) = sup 0 ≤ t ≤ T ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ u ( t ) ‖ 2 + ( 1 + t ) ( a − 1 2 ) ( β + 1 ) + γ ‖ ∇ u ( t ) ‖ 2 + ( 1 + t ) γ − β ‖ ∂ t u ( t ) ‖ 2

when a > 0 (i.e., m < 2 ), and

‖ u ‖ X ( T ) = sup 0 ≤ t ≤ T ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ u ( t ) ‖ 2 + ( 1 + t ) ( a − 1 2 ) ( β + 1 ) + γ ‖ ∇ u ( t ) ‖ 2 + ( 1 + t ) γ − β ( ln ( 2 + t ) ) − 1 ‖ ∂ t u ( t ) ‖ 2

when a = 0 (i.e., m = 2 ), for any v ∈ X ( T ) .

• Φ : B R ( T ) ⟶ B R ( T ) . Let u ∈ B R ( T ) .

Case of a > 0 .

Estimation of ‖ Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2 and the fact that γ − β < 1 , we have

(30) ‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ

and

‖ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

By Gagliardo-Nirenberg inequality, we obtain

‖ u ( s ) ‖ j p p ≤ C ‖ u ( s ) ‖ 2 p ( 1 − θ ( j p ) ) ‖ ∇ u ( s ) ‖ 2 p θ ( j p ) ≤ C ( 1 + s ) − β j ⋆ ‖ u ‖ X ( T ) p , j = 2 , m ,

where

β j ⋆ = p n 2 m − 1 ( β + 1 ) + γ − n ( β + 1 ) 2 j , j = 2 , m .

As m < 2 , we can see that β 2 ⋆ > β m ⋆ and

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ⋆ ‖ u ‖ X ( T ) p ,

whereupon

‖ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m ⋆ d s d τ .

Note that β m ⋆ > 1 ⇔ p > p 2 * , which implies, using the fact that max { a , γ − β } < 1 and Lemma 1, that

(31) ‖ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

By equations (30) and (31), we obtain

(32) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) ‖ 2 ̲ : By γ − β < 1 and Theorems 1 and 2, we obtain

(33) ‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( a + 1 2 ) ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 2 − a ) ( β + 1 ) − γ

and

‖ ∇ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

Similar to the estimation of ‖ Φ ( u ) ( t ) ‖ 2 , we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ⋆ ‖ u ‖ X ( T ) p ,

therefore

‖ ∇ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 2 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m ⋆ d s d τ .

Note that β m ⋆ > 1 ⇔ p > p 2 * , which implies, using the fact that max { a + 1 2 , γ − β } < 1 and Lemma 1, that

(34) ‖ ∇ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 2 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

By equations (33) and (34), we obtain

(35) ( 1 + t ) ( a − 1 2 ) ( β + 1 ) + γ ‖ ∇ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and the fact that a > 0 , γ − β < 1 , we have

(36) ‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) − 1 ≤ C ε 0 ( 1 + t ) β − γ

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

As above,

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ⋆ ‖ u ‖ X ( T ) p ,

therefore

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − a − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m ⋆ d s d τ .

Note that β m ⋆ > 1 ⇔ p > p 2 * , which implies, using the fact that max { a + 1 , γ − β } = a + 1 > 1 and Lemma 1, that

(37) ‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ ‖ u ‖ X ( T ) p .

By equations (36) and (37), we obtain

(38) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Summing up equations (32), (35), and (38) and using the fact that u ∈ B R ( T ) , we conclude that

‖ Φ ( u ) ‖ X ( T ) ≤ C ε 0 + C ‖ u ‖ X ( T ) p ≤ C ε 0 + C R p .

At this stage, letting first R > 0 such that C R p − 1 ≤ 1 , and then taking 0 < ε 0 ≪ 1 such that R ≥ C ε 0 , we arrive at

‖ Φ ( u ) ‖ X ( T ) ≤ 2 R ,

i.e., Φ ( u ) ∈ B R ( T ) .

Case of a = 0 .

Estimation of ‖ Φ ( u ) ( t ) ‖ 2 ̲ : Using same calculations as in the last case ( a > 0 ), we obtain

‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − a ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ

and

‖ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

Therefore,

(39) ( 1 + t ) ( a − 1 ) ( β + 1 ) + γ ‖ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) ‖ 2 ̲ : As argued in the last case ( a > 0 ), we have

‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( a + 1 2 ) ( β + 1 ) ≤ C ε 0 ( 1 + t ) ( 1 2 − a ) ( β + 1 ) − γ

and

‖ ∇ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) ( 1 2 − a ) ( β + 1 ) − γ ‖ u ‖ X ( T ) p .

Therefore, we have

(40) ( 1 + t ) ( a − 1 2 ) ( β + 1 ) + γ ‖ ∇ Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and the fact that a = 0 , γ − β < 1 , we have

(41) ‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − 1 ≤ C ε 0 ( 1 + t ) β − γ ln ( 2 + t )

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

In addition,

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β m ⋆ ‖ u ‖ X ( T ) p

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β m ⋆ d s d τ .

As β m ⋆ > 1 ⇔ p > p 2 * and using the fact that max { 1 , γ − β } = 1 and Lemma 1, we have

(42) ‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ ln ( 2 + t ) ‖ u ‖ X ( T ) p .

By equations (41) and (42), we obtain

(43) ( 1 + t ) γ − β ( ln ( 2 + t ) ) − 1 ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ≤ C ε 0 + C ‖ u ‖ X ( T ) p .

Summing up equations (39), (40), and (43) and using the fact that u ∈ B R ( T ) , we conclude that

‖ Φ ( u ) ‖ X ( T ) ≤ C ε 0 + C ‖ u ‖ X ( T ) p ≤ C ε 0 + C R p .

At this stage, choosing first R > 0 such that C R p − 1 ≤ 1 , and then taking 0 < ε 0 ≪ 1 such that R ≥ C ε 0 , we arrive at

‖ Φ ( u ) ‖ X ( T ) ≤ 2 R ,

i.e. Φ ( u ) ∈ B R ( T ) .

Using the calculations similar to the ones used in the proof of Theorem 3, we can easily deduce that Φ is a contraction. Therefore, by the Banach fixed point theorem, there exists a unique mild solution u ∈ X ( T ) of problem ( P ) . This completes the proof of Theorem 4.□

Proof of Theorem 5

Let us introduce, for T > 0 , the same space of energy solutions X ( T ) equipped with the norm

‖ u ‖ X ( T ) = sup 0 ≤ t ≤ T { ( 1 + t ) γ ( ln ( 2 + t ) ) − 1 ‖ u ( t ) ‖ 2 + ( 1 + t ) γ ‖ ∇ u ( t ) ‖ 2 + ( 1 + t ) γ − β ‖ ∂ t u ( t ) ‖ 2 }

for any v ∈ X ( T ) .

•   Φ : B R ( T ) ⟶ B R ( T ) . Let u ∈ B R ( T ) .

Estimation of ‖ Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and γ − β < 1 , we obtain

(44) ‖ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( β + 1 ) ≤ C ε 0 ( 1 + t ) − γ ln ( 2 + t )

and

‖ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

By Gagliardo-Nirenberg inequality, we have

‖ u ( s ) ‖ j p p ≤ C ‖ u ( s ) ‖ 2 p ( 1 − θ ( j p ) ) ‖ ∇ u ( s ) ‖ 2 p θ ( j p ) , j = 2 , m .

As a = 1 , we have

‖ u ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ‖ u ‖ X ( T ) ≤ C ( 1 + t ) − γ + δ ⋆ ‖ u ‖ X ( T )

for δ ⋆ ≪ 1 to be chosen later. Therefore, we obtain

‖ u ( s ) ‖ j p p ≤ C ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( j p ) ) ‖ u ‖ X ( T ) p , j = 2 , m ,

where

β ⋆ = p γ , j = 2 , m .

As m < 2 , we have θ ( m p ) < θ ( 2 p ) , and then,

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) ‖ u ‖ X ( T ) p .

Hence, we have

‖ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 1 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) d s d τ .

Note that β ⋆ > 1 ⇔ p > 1 γ , so by choosing δ ⋆ ≪ 1 , namely δ ⋆ < ( β ⋆ − 1 ) ⁄ ( p ( 1 − θ ( m p ) ) ) , we obtain − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) > 1 , which implies, using the fact that max { a , γ − β } = 1 and Lemma 1, that

(45) ‖ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ln ( 2 + t ) ‖ u ‖ X ( T ) p .

Note that θ ( m p ) < θ ( 2 p ) ≤ 1 , i.e., θ ( m p ) < 1 . By equations (44) and (45), we obtain

(46) ( 1 + t ) γ ( ln ( 2 + t ) ) − 1 ‖ Φ ( u ) ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∇ Φ ( u ) ( t ) ‖ 2 ̲ : By Theorems 1 and 2, and γ − β < 1 , we obtain

(47) ‖ ∇ u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − 3 2 ( β + 1 ) ≤ C ε 0 ( 1 + t ) − γ

and

‖ ∇ u n l ( t ) ‖ 2 ≤ C ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 3 2 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

As in the estimation of ‖ Φ ( u ) ( t ) ‖ 2 , we have

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) ‖ u ‖ X ( T ) p ,

which yields

‖ ∇ u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 3 2 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) d s d τ .

Note that β ⋆ > 1 ⇔ p > 1 γ , so by the same choice of δ ⋆ , we arrive at − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) > 1 , which implies, using the fact that max { 3 2 , γ − β } = 3 2 > 1 and Lemma 1, that

(48) ‖ ∇ u n l ( t ) ‖ 2 ≤ C ( 1 + t ) − γ ‖ u ‖ X ( T ) p .

By equations (47) and (48), we obtain

(49) ( 1 + t ) γ ‖ ∇ Φ ( u ) ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p .

Estimation of ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ̲ : Using Theorems 1 and 2, and γ − β < 1 , we have

(50) ‖ ∂ t u lin ( t ) ‖ 2 ≤ C I 0 ( 1 + t ) − ( β + 1 ) − 1 ≤ C ε 0 ( 1 + t ) β − γ

and

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 2 ∫ 0 τ ( τ − s ) − γ ( ‖ u ( s ) ‖ m p p + ‖ u ( s ) ‖ 2 p p ) d s d τ .

In addition,

‖ u ( s ) ‖ m p p , ‖ u ( s ) ‖ 2 p p ≤ C ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) ‖ u ‖ X ( T ) p ,

whereupon

‖ ∂ t u n l ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p ( 1 + t ) β ∫ 0 t ( 1 + τ ) β ( 1 + B ( t , τ ) ) − 2 ∫ 0 τ ( τ − s ) − γ ( 1 + s ) − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) d s d τ .

Note that β ⋆ > 1 ⇔ p > 1 γ . Then, using the fact that − β ⋆ + δ ⋆ p ( 1 − θ ( m p ) ) > 1 , max { 2 , γ − β } = 2 > 1 and Lemma 1, we find that

(51) ‖ ∂ t u n l ( t ) ‖ 2 ≤ C ( 1 + t ) β − γ ‖ u ‖ X ( T ) p .

By equations (50) and (51), we obtain

(52) ( 1 + t ) γ − β ‖ ∂ t Φ ( u ) ( t ) ‖ 2 ≤ C ‖ u ‖ X ( T ) p .

Summing up equations (46), (49), (52) and using the fact that u ∈ B R ( T ) , we conclude that

‖ Φ ( u ) ‖ X ( T ) ≤ C ε 0 + C ‖ u ‖ X ( T ) p ≤ C ε 0 + C R p .

Choosing R > 0 such that C R p − 1 ≤ 1 , and then taking 0 < ε 0 ≪ 1 such that R ≥ C ε 0 , we arrive at

‖ Φ ( u ) ‖ X ( T ) ≤ 2 R ,

i.e., Φ ( u ) ∈ B R ( T ) . Using the similar calculations as in the proof of Theorem 3, we can easily prove that Φ is a contraction. Thus, by the Banach fixed point theorem, there exists a unique mild solution u ∈ X ( T ) of problem ( P ) . This completes the proof of Theorem 5.□

Acknowledgments

Dr. M. Kirane thanks Khalifa University for its support.

  1. Funding information: This research work was funded by Institutional Fund Projects under grant no. (IFPIP: 367-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

  2. Conflict of interest: The authors state that there is no conflict of interest.

References

[1] M. D’Abbicco, A wave equation with structural damping and nonlinear memory, J. Nonlinear Differ. Equ. Appl. 21 (2014), 751–773. 10.1007/s00030-014-0265-2Search in Google Scholar

[2] M. D’Abbicco, The influence of a nonlinear memory on the damped wave equation, J. Nonlinear Analysis 95 (2014), 130–145. 10.1016/j.na.2013.09.006Search in Google Scholar

[3] M. D’Abbicco, S. Lucente, and M. Reissig, Semilinear wave equations with effective damping, Chin. Ann. Math. Serie B 34 (2013), no. 3, 345–380. 10.1007/s11401-013-0773-0Search in Google Scholar

[4] A. Z. Fino, Critical exponent for damped wave equations with nonlinear memory, J. Nonlinear Analysis 74 (2011), 5495–5505. 10.1016/j.na.2011.01.039Search in Google Scholar

[5] H. Fujita, On the blowing up of solutions of the problem for ut=Δu+u1+α, J. Fac. Sci. Univ. Tokyo 13 (1966), 109–124. Search in Google Scholar

[6] M. Ikawa, Hyperbolic Partial Differential Equations and Wave Phenomena, American Mathematical Society, Providence, RI, 2000. 10.1090/mmono/189Search in Google Scholar

[7] R. Ikehata and K. Tanizawa, Global existence for solutions for semilinear damped wave equation in RN with noncompactly supported initial data, Nonlinear Analysis 61 (2005), 1189–1208. 10.1016/j.na.2005.01.097Search in Google Scholar

[8] T. H. Kaddour and M. Reissig, Global well-posedness for effectively damped wave models with nonlinear memory, Commun. Pure Appl. Anal. 20 (2021), no. 5, 2039–2064. 10.3934/cpaa.2020239Search in Google Scholar

[9] M. Kirane and M. Qafsaoui, Fujita’s exponent for a semilinear wave equation with linear damping, Adv Nonlinear Studies 2, (2002) no. 1, 41–49. 10.1515/ans-2002-0103Search in Google Scholar

[10] T.-T. Li and Y. Zhou, Breakdown of solutions to □u+ut=∣u∣1+α, Discrete Contin. Dyn. Syst. 1 (1995), 503–520. 10.3934/dcds.1995.1.503Search in Google Scholar

[11] J. Lin, K. Nishihara, and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst. 32 (2012), 4307–4320. 10.3934/dcds.2012.32.4307Search in Google Scholar

[12] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327–343. 10.3836/tjm/1327931389Search in Google Scholar

[13] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, 1987. Search in Google Scholar

[14] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001), 464–489. 10.1006/jdeq.2000.3933Search in Google Scholar

[15] Y. Wakasugi, On the diffusive structure for the damped wave equation with variable coefficients, Doctoral thesis, Osaka University, Osaka, Japan, 2014. Search in Google Scholar

[16] Q. S. Zhang, A blow up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, 333 (2001), no. 2, 109–114. 10.1016/S0764-4442(01)01999-1Search in Google Scholar
Received: 2023-02-18
Revised: 2023-09-01
Accepted: 2023-09-27
Published Online: 2023-11-18

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
https://doi.org/10.1515/anona-2023-0111
Keywords for this article
nonlinear damped wave equation; global existence
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2023-0111/html
Scroll to top button