Journal of Function Spaces
Volume 2020 (2020), Article ID 4867186, 7 pages
https://doi.org/10.1155/2020/4867186

Research Article

On Local Weak Solutions for Fractional in Time SOBOLEV-Type Inequalities

Mohaemd Jleli and Bessem Samet

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Correspondence should be addressed to Bessem Samet; bsamet@ksu.edu.sa

Received 8 July 2020; Revised 24 August 2020; Accepted 10 September 2020; Published 23 September 2020

Academic Editor: Zoran Mitrovic

Copyright © 2020 Mohaemd Jleli and Bessem Samet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider two fractional in time nonlinear Sobolev-type inequalities involving potential terms, where the fractional derivatives are defined in the sense of Caputo. For both problems, we study the existence and nonexistence of nontrivial local weak solutions. Namely, we show that there exists a critical exponent according to which we have existence or nonexistence.

1. Introduction and Main Results

We consider the fractional in time Sobolev-type inequalities

and

where , , and . Here, and is the derivative of fractional order in the sense of Caputo. Namely, we are concerned with the existence and nonexistence of nontrivial local weak solutions to problems (1) and (2). We shall establish that there exists a critical exponent that depends on and such that if p ∈ (1, ), then problems (1) and (2) admit no nontrivial local weak solutions (i.e., we have an instantaneous blow-up), while if , then the considered problems admit local solutions for some initial values. In the proofs of our results, we use the test function method with some integral estimates. For more details about the test function method and its applications to partial differential equations, we refer to [1–3] and the references therein.

The absence of solutions (complete blow-up phenomenon) was observed in [4] for the following elliptic inequality with a singular potential term

where is a smooth bounded domain in containing 0. In the same reference, an instantaneous blow-up result was obtained for the parabolic analogue of (3), namely

Notice that the method in [4] is based on comparison principles. In [5], using the test function method and avoiding the maximum principle, instantaneous blow-up results were obtained for certain classes of elliptic and parabolic inequalities including as special cases (3) and (4). For more results on instantaneous blow-up for nonlinear evolution equations, we refer to [6–9] and the references therein.

The investigation of instantaneous blow-up for linear Sobolev-type equations was first considered in [10]. Namely, the following problem was studied

In the limit case and , (1) and (2) (with equalities instead of inequalities), reduce, respectively, to

The nonexistence of local weak solutions to (6) and (7) was considered in [11] by the use of the test function method. When , it was proved that for all , (6) and (7) admit no nontrivial local weak solutions. If , it was shown that if , then (6) and (7) admit no nontrivial local weak solutions, while if , then local solutions exist. Our aim in this paper is to study the instantaneous blow-up for the fractional in time versions of (6) and (7) with the potential term .

For existence results for stationary problems involving potential terms, see for example [12] and the references therein.

Notice that the study of fractional in time Sobolev-type equations was first considered in [13], where the nonexistence of global weak solutions was investigated.

Before mentioning our main results, let us define the meaning of solutions to (1) and (2).

Let . Given and , , we define the fractional integral operators

and

Given , , one has (see e.g., [14, 15])

For , the derivatives of fractional orders and in the Caputo sense are defined, respectively, by

Using the above notions and property (10), we define local weak solutions to problem (1) as follows.

Definition 1. Let . We say that is a local weak solution to (1), if there exists and satisfies

for all , , with . Here, is the product set . Local weak solutions to problem (2) are defined as follows.

Definition 2. Let . We say that is a local weak solution to (2), if there exists such that and satisfies

for all , 0, with and .

Now, we state our results. We first define the (critical) exponent

Theorem 3. Let , and .

(i)Ifand, then problem (1) admits no nontrivial local weak solution(ii)Ifand, then problem (1) admits local solutions for some

Theorem 4. Let , , and .

(i)Ifand, then problem (2) admits no nontrivial local weak solution(ii)Ifand, then problem (2) admits local solutions for someand

The next section contains some preliminary estimates that will be useful in the proofs of our results. Section 3 is devoted to the proofs of Theorems 3 and 4.

2. Preliminaries

For , let

and

where (i.e., sufficiently large) is a natural number and satisfies

Let us introduce the function

Clearly, one has .

Lemma 5. The function defined by (15) satisfies the following properties:

where .

Proof. One has

Taking , the above integral reduces to

which proves (19). Next, (20) and (21) follow by differentiating (19).

Lemma 6. For sufficiently large , one has

Here, is a constant (independent on ).

Proof. Using(16) and (17), one obtains

On the other hand, an elementary calculation shows that

Here and below, is a constant independent on , whose value may change from line to line. Hence, one deduces that

which proves the desired result.

The following result is obvious.

Lemma 7. For , one has

Using (20), one obtains easily the following result.

Lemma 8. For , one has

where .

Using (21), the following result follows.

Lemma 9. For , one has

where .

3. Proofs of the Main Results

Proof of Theorem 10.

(i)Suppose that is a nontrivial local wessak solution to (1) for some fixed . Then, using (12) with is the function defined by (18), one obtains

Next, using ε-Young inequality with , one obtains

Similarly, one has

It follows from (31), (32), and (33) that

On the other hand, by (18), one has

Hence, using Lemmas 6 and 7, for sufficiently large , one obtains

Again, by (18), one has

Therefore, using Lemma 6 and Lemma 8 with , for sufficiently large , one obtains

On the other hand,

Hence, using (19) (with ) and (26), one deduces that

Notice that since , by (17), one deduces from the above estimate that

Next, it follows from (34), (36), (38), and (40) that

Notice that since , then . Hence, passing to the infimum limit as in (42), using (17), (41), and Fatou’s lemma, one deduces that

which contradicts the fact that is nontrivial.

(ii)Let

Notice that since , the set of values of satisfying (44) is nonempty. Consider the function

An elementary calculation shows that

Hence, using (44), (45), and (46), one obtains

This shows that

is a global solution (so local solution) to (1) with .

Proof of Theorem 11.

(i)Suppose that is a nontrivial local weak solution to (2) for some fixed . Then, using (13) with is the function defined by (18), and one obtains

Notice that by (20) (with ), one has . Hence, by Definition 2, the choice of the test function defined by (18) is allowed. Next, following the same arguments used in the proof of part (i) of Theorem 3, by the use of ε-Young inequality, one obtains

where

Notice that by (18), one has

Hence, using Lemma 6 and Lemma 9 with , for sufficiently large , one deduces that

Furthermore, by (19) and (20) (with ), one has

Next, using (36), (38), (51), (54), (55), and (56), for sufficiently large , one obtains

Notice that by (26), since , one has

Therefore, passing to the infimum limit as in (57) and using Fatou’s lemma, since , it holds that

which contradicts the fact that is nontrivial.

(ii)From the proof of part (ii) of Theorem 3, one deduces that the function , where is defined by (46), is a global solution (so local solution) to (2) with and

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-237.

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