Semi-Linear Fractional σ-Evolution Equations with Nonlinear Memory

KAINANE MEZADEK Abdelatif

1 Université Hassiba Benbouali de Chlef,Faculté des Sciences Exactes et Informatique,Départment de Mathematiques,P.O.Box 151,02000 Ouled Fares Chlef,Algeria.

2 Laboratoire d’Analyse et Contr?le des Equations aux Dérivées Partielles,P.O.Box 89,22000 Sidi Bel Abbes,Algeria.

Abstract. In this paper we study the local or global (in time) existence of small data solutions to semi-linear fractional σ?evolution equations with nonlinear memory.Our main goals is to explain on the one hand the influence of the memory term and on the other hand the influence of higher regularity of the data on qualitative properties of solutions.

Key Words: Fractional equations; σ-evolution equations; global in time existence; small data solutions;nonlinear memory.

1 Introduction

Fractional integrals and Fractional derivatives have applications in many fields including engineering,science,finance,applied mathematics,bio-engineering,radiative transfer, neutron transport, and the kinetic theory of gases, see, e.g. [1-3] to illustrate some applications. We refer also to the references [4, 5] for an introduction on the theory of fractional derivatives.

This note is devoted to the Cauchy problem for the semi-linear fractionalσ-evolution equations with nonlinear memory. We are interested to the existence of solutions to the following Cauchy problem for the semi-linear fractionalσ-evolution equations with nonlinear memory

whereα∈(0,1),σ≥1,μ∈(0,1),p＞1,(t,x)∈[0,∞)×Rn,with

anddenotes the fractional Riemann-Liouville derivative and the fractional Riemann-Liouville integral respectively offin[0,t]and Γ is the Euler Gamma function.Our main goal is to understand on the one hand the improving influence of the nonlinear memory and on the other hand the influence of higher regularity of the datau0on the solvability behavior.

Remark 1.1.This problem has been studied in the caseα=1 by several authors.We refere to reference[6] for the caseσ=2 and to references[7,8] for the caseσ=1 with damped term.Also we refere to reference[9]for the caseσ=1 with structural damping.

In [10], Kainane and Reissig studied the following Cauchy problem for semi-linear fractionalσ-evolution equations with power non-linearity

whereα∈(0,1),σ≥1. The authors proved the following results.

Proposition 1.1.Let us assume0＜α＜1,σ≥1and r≥1. We assume thatMoreover,the exponent p satisfies the condition

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.2). Moreover, the solution satisfies the following estimate for any t≥0and for all sufficiently small δ＞0:

where

The constant C is independent of u0.

Proposition 1.2.Let us assume0＜α＜1,σ≥1,1＜r＜∞and γ≥0. We assume thatThe exponent p satisfies the condition

where

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.2). The solution satisfies the following estimate for any t≥0and for all sufficiently small δ＞0:

where

Moreover,the solution satisfies the estimate

The constants C are independent of u0.

Proposition 1.3.Let us assumeWe assume thatMoreover,the exponent p satisfies the condition

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.2). Moreover,the solution satisfies the following estimate for any t≥0:

where

The constant C is independent of u0.

Proposition 1.4.Let us assumeand γ≥0. We assume thatMoreover,the exponent p satisfies the condition

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.2). The solution satisfies the following estimate for any t≥0:

where

Moreover,the solution satisfies the estimate

The constants C are independent of u0.

As in[10],by the assumptionut(0,x)=0,the Cauchy problem(1.1)may be written in the form of a Cauchy problem for an integro-differential equation

where

A solution to(1.1)is defined as a solution of(1.3).

Our results of global (in time)existence of small data Sobolev solutions are given in the next sections.

In the following sections we use the notationwitch means that there exists a constantC≥0 such thatf≤Cg.

Our results of global(in time)existence of small data weak solutions are given in the next sections.

2 Main results in the case

Theorem 2.1.Let us assume0 ＜α＜1, α＜μ＜1, σ≥1and r≥1. We assume that Moreover,the exponent p satisfies the condition

where

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.1). The solution satisfies the following decay estimate for any t≥0and for all sufficiently small δ＞0

where

The constant C is independent of u0.

Theorem 2.2.Let us assume0＜α＜1,α＜μ＜1,σ≥1,1＜r＜∞and γ≥0

Surely you do not doubt the existence of a future life? exclaimed the young wife. It seemed as if one of the first shadowspassed over her sunny thoughts.

where

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.1). The solution satisfies the following decay estimate for any t≥0and for all sufficiently small δ＞0

where

Moreover,the solution satisfies the estimate

The constant C is independent of u0.

3 Main results in the case

Theorem 3.1.Let us assumeWeassume thatMoreover,the exponent p satisfies the condition

where

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.1). The solution satisfies the following decay estimate for any t≥0

where

The constant C is independent of u0.

Theorem 3.2.Let us assumeandγ≥0. We assume that. Moreover,the exponent p satisfies the condition

where

Then there exists a positive constant ε such that for any data

we have a uniquely determined global(in time)weak solution

to the Cauchy problem(1.1). The solution satisfies the following decay estimate for any t≥0

where

Moreover,the solution satisfies the estimate

The constant C is independent of u0.

Remark 3.1.The nonlinear termF(t,u)in(1.4)may be written as

where Γ is the Euler Gamma function,andis the fractional Riemann-Liouville integral of |u|pin [0,t]. Therefore, it is reasonable to expect relations with the case of a power nonlinearityF(u)=|u|pasμtends to 1.

4 Some preliminaries

The Cauchy problem (1.1) withσ≥1 can be formally converted to an integral equation and its solution is given by

with

where {Gα,σ(t)}t≥0denotes the semigroup of operators which is defined via Fourier transform by

Heredenotes the Mittag-Leffler function(see[10],Section 7.2).

A representation of solutions of the linear integro-differential equation associated to(1.3)or(1.1)withσ≥1(and without the termF(t,u))is given by(see[10]or[11]).

5 Lr-Lq estimates for solutions to linear Cauchy problems

The linear estimate of the Cauchy problem is given by the following proposition

Proposition 5.1.(see [10]) Let u0∈Lr(n)∩L∞(n), n≥1, r≥1and α∈(0,1). Then the function

satisfies the following estimate for any fixed δ＞0small enough:

where

≥1,1＜r＜∞,γ≥0,and α∈(0,1). Then the function

satisfies the following estimates:

Corollary 5.1.([10])Let u0∈Lr(n)∩L∞(n),n≥1,r≥1and α∈(0,1). Then the function

belongs to

≥1,1＜r＜∞,γ≥0and α∈(0,1). Then the function

belongs to

The next result contains even the continuity property with respect to the time variable.

Proposition 5.2.([10])Let u0∈Lr(n)∩L∞(n),n≥1,r≥1and α∈(0,1). Then the function

belongs to

≥1,1＜r＜∞,γ≥0and α∈(0,1). Then the function

belongs to

6 Proofs of the main results

6.1 Proof of Theorem 2.1

For anyandδ∈(0,1)is sufficiently small,there exists a parametersuch that

We define the space

with the norm

For anyu∈X(T),we consider the operator

We shall prove that

After proving (6.2) and (6.3) we may conclude the global (in time) existence result in Theorem 2.1. For the proof of (6.2), after taking into consideration the linear estimates(5.1),we have

Consequently,

for anyq∈[r,∞]and due toThanks to(5.1)and(6.5)we can estimate

where

We are interested to estimate the functionIq(t) in (6.6). For this we apply Lemma 7.1 from Appendix.We notice thatif and only if

Consequently,by using Lemma 7.1 we may estimate as follows:

thanks to the fact thatandα＜μ＜1. Therefore(6.5)gives

Finally,it remains to show(6.3). Letq∈[r,∞). By H?lder’s inequality,foru,v∈X(T),and ifp′denotes the conjugate top,we have

Moreover,we have

Hence,

We deduce that

Notice thatp＞pα,μ,σ,r,δfor allδ＞0 if and only ifp＞pα,μ,σ,r.

Remark 6.1.All the estimates (6.2) and (6.3) are uniformly with respect toT∈(0,∞) ifp＞pα,μ,σ,r.

From(6.2)it follows thatPmapsX(T)into itself for allTand for small data.By standard contraction arguments(see[12])the estimates(6.2)and(6.3)lead to the existence of unique solution tou=Puand,consequently,to(1.1). Since all constants are independent ofTwe letTtend to ∞and we obtain a global (in time) existence result for small data solutions to(1.1).

Finally, let us discuss the continuity of the solution with respect tot. The solution satisfies the operator equation

The above estimates forNα,σ(u)and the integral termimply for allT＞0

Proposition 5.2 gives

Consequently,

what we wanted to have.

If the data are large,then instead we get forp＞1 the estimates

whereC(T)tends to 0 forT→+0. For this reason we can have for general(large)data a global(in time)existence result of weak solutions only.The proof is complete.

6.2 Proof of Theorem 2.2

We define the solution space

with the norm

whereis defined as in Subsection 6.1. For anyu∈X(T),we consider the operator

We shall prove that

After proving (6.9) and (6.10) we may conclude the global (in time) existence result in Theorem 2.2. For the proof of(6.9),after taking account of the estimates(5.1)and(5.2)we have

Moreover,we have

As in Section 6.1 we deduce

if and only if

Now let us turn to the desired estimate of the normWe need to estimate the normApplying Proposition 7.2 we obtain forp＞max{2;γ}

Then

where

If

then

We remark thatThen we deduce that

if and only if

Finally,we have to show(6.10). From Section 6.1 we get the estimate

for allt∈[0,T]andq∈[r,∞]. It remains to prove

From the above considerations it is sufficient to prove that

By using the integral representation

whereF(u)=u|u|p?2,we obtain

Applying the fractional Leibniz formula from Proposition 7.4 to estimate a product inwe get

We apply again Proposition 7.2 to estimate the term inside of the integral.In this way we obtain

Then

Hence,

We deduce that

Notice thatfor allδ＞0 if and only ifp＞pα,μ,σ,r.

If the data are large,then instead we get forp＞2 the estimates

whereC(T)tends to 0 forT→+0. For this reason we can have for general(large)data a local(in time)existence result of weak solutions only.

As at the end of the proof of Theorem 2.1 we verify that the solutionubelongs even to

The proof is complete.

6.3 Proof of Theorem 3.1

Hence, we can choose a positiveδsuch that there does not exist any∈[r,∞] which satisfies(6.1). For this reason,

We define the space

with the norm

We shall prove that

After proving (6.19) and (6.20) we may conclude the global (in time) existence result in Theorem 3.1. For the proof of(6.19), after taking into consideration the linear estimates(5.1),we have

Consequently,

for anyq∈[r,∞]and due toThanks to(5.1)and(6.5)we can estimate

where

We are interested to estimate the functionIq(t)in(6.6). For this we apply Lemma 7.1. We notice thatif and only if

where

thanks to the fact thatTherefore(6.22)gives

Finally,the proof of(6.20)is similar as in Section 6.1.

As at the end of the proof of Theorem 2.1 we verify that the solutionubelongs even to

The proof is complete.

6.4 Proof of Theorem 3.2

We define the solution space

with the norm

whereFor anyu∈X(T),we consider the operator

We shall prove that

After proving (6.24) and (6.10) we may conclude the global (in time) existence result in Theorem 3.2. For the proof of(6.24), after taking account of the estimates(5.1) and(5.2)we have

Moreover,we have

As in Section 6.3 we deduce

if and only if

Now let us turn to the desired estimate of the normWe need to estimate the normApplying Proposition 7.2 we obtain forp＞max{2;γ}

Then

where

If

We remark thatThen we deduce that

if and only if

Finally,the proof of(6.25)is similar as in Section 6.2.

As at the end of the proof of Theorem 2.1 we verify that the solutionubelongs even to

This completes the proof.

7 Appendix

7.1 Results from Harmonic Analysis

We recall some results from Harmonic Analysis(cf. with[13]).

Proposition 7.1.Let r∈(1,∞),p＞1and σ∈(0,p). Let Q(u)denote one of the functions|u|p,±u|u|p?1. Then the following inequality holds:

≥0and1＜q＜∞the fractional Sobolev spaces or Bessel potential spaces

Moreover,〈D〉γ stands for the pseudo-differential operator with symboland it is defined by

Proposition 7.2.Let r∈(1,∞),p＞1and σ∈(0,p). Let Q(u)denote one of the functions|u|p,±u|u|p?1. Then the following inequality holds:

for anywhere

Here|D|γ stands for the pseudo-differential operator with symboland it is defined by|D|γu=

Proposition 7.3.Let r∈(1,∞)and σ＞0. Then the following inequality holds:

for any

Finally let us state the corresponding inequality in homogeneous spacesFor the proof it is possible to follow the same strategy as in the proof of Proposition 7.2.

Proposition 7.4(Fractional Leibniz formula).Let r∈(1,∞)and σ＞0. Then the following inequality holds:

for any

We also recall the following lemma from[14].

Lemma 7.1.Suppose that θ∈[0,1),a≥0and b≥0. Then there exists a constant C=C(a,b,θ)＞0such that for all t＞0the following estimate holds:

Acknowledgments

The research of this article is supported by the DAAD, Erasmus+ Project between the Hassiba Benbouali University of Chlef(Algeria)and TU Bergakademie Freiberg,2015-1-DE01-KA107-002026,during the stay of the author at Technical University Bergakademie Freiberg within the period April to July 2017. The author expresses a sincere thankfulness to Prof.Michael Reissig for numerous discussions and the staff of the Institute of Applied Analysis for their hospitality. The author thank the reviewer for his/her comments and suggestions.