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Mohaemd Jleli, Bessem Samet, "On Local Weak Solutions for Fractional in Time SOBOLEV-Type Inequalities", Journal of Function Spaces, vol. 2020, Article ID 4867186, 7 pages, 2020. https://doi.org/10.1155/2020/4867186
On Local Weak Solutions for Fractional in Time SOBOLEV-Type Inequalities
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  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  is based on comparison principles. In , 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 . Namely, the following problem was studied
The nonexistence of local weak solutions to (6) and (7) was considered in  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  and the references therein.
Notice that the study of fractional in time Sobolev-type equations was first considered in , where the nonexistence of global weak solutions was investigated.
Let . Given and , , we define the fractional integral operators and
For , the derivatives of fractional orders and in the Caputo sense are defined, respectively, by
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
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
Lemma 6. For sufficiently large , one has
Here, is a constant (independent on ).
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
Next, using ε-Young inequality with , one obtains
Similarly, one has
On the other hand, by (18), one has
Again, by (18), one has
Notice that since , by (17), one deduces from the above estimate that
Notice that since , the set of values of satisfying (44) is nonempty. Consider the function
An elementary calculation shows that
This shows that is a global solution (so local solution) to (1) with .
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
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
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.
All authors contributed equally to this work.
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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