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

where N ≥ 1, p > 1, and ρ > −2. Here, σ ∈ ð0, 1Þ and ∂t , λ ∈ fσ, σ + 1g is the derivative of fractional order λ 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 pc > 1 that depends on N and ρ such that if p ∈ (1, pc), then problems (1) and (2) admit no nontrivial local weak solutions (i.e., we have an instantaneous blow-up), while if p ∈ ðpc,∞Þ, 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


Introduction and Main Results
We consider the fractional in time Sobolev-type inequalities and where N ≥ 1, p > 1, and ρ > −2. Here, σ ∈ ð0, 1Þ and ∂ λ t , λ ∈ fσ, σ + 1g is the derivative of fractional order λ 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 p c > 1 that depends on N and ρ such that if p ∈ (1, p c ), then problems (1) and (2) admit no nontrivial local weak solutions (i.e., we have an instantaneous blow-up), while if p ∈ ðp c ,∞Þ, 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][2][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 Ω is a smooth bounded domain in R N 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][7][8][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 σ = 1 and ρ = 0, (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 N ∈ f1, 2g, it was proved that for all p > 1, (6) and (7) admit no nontrivial local weak solutions. If N ≥ 3, it was shown that if 1 < p ≤ N/N − 2, then (6) and (7) admit no nontrivial local weak solutions, while if P > N/N − 2, 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 VðzÞ = ð1 + jzj 2 Þ ρ/2 .
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).
for all ξ ∈ C∞ðΛ T,N Þ, ξ ≥ 0, with supp z ðξÞ ⊂ ⊂R N . Here, Λ T,N is the product set ½0, T × R N . Local weak solutions to problem (2) are defined as follows.
Proof. One has Taking τ = T − ς/T − t, the above integral reduces to Here, C > 0 is a constant (independent on S).
Proof. Using (16) and (17), one obtains On the other hand, an elementary calculation shows that Here and below, C > 0 is a constant independent on S, whose value may change from line to line. Hence, one deduces that which proves the desired result.
The following result is obvious.

Journal of Function Spaces
Using (20), one obtains easily the following result.

Proofs of the Main Results
Proof of Theorem 10.