On the critical behavior for time-fractional pseudo-parabolic-type equations with combined nonlinearities

We are concerned with the existence and nonexistence of global weak solutions for a certain class of time-fractional inhomogeneous pseudo-parabolic-type equations involving a nonlinearity of the form | u | p + ι |∇ u | q , where p , q > 1, and ι ≥ 0 is a constant. The cases ι = 0 and ι > 0 are discussed separately. For each case, the critical exponent in the Fujita sense is obtained. We point out two interesting phenomena. First, the obtained critical exponents are independent of the fractional orders of the time derivative. Secondly, in the case ι > 0, we show that the gradient term induces a discontinuity phenomenon of the critical exponent. MSC: 35K70; 35B33; 35B44; 34K37


Introduction
In this paper, we consider equations driven by time-fractional pseudo-parabolic-type operators of the form u → ∂ α t uk ∂ β t u for functions u = u(t, x) defined on (0, ∞) × R N , where N ≥ 1, k > 0, and 0 < α, β < 1. Here, for ρ ∈ {α, β}, ∂ ρ t denotes the time-Caputo fractional derivative of order ρ (see Sect. 2). We study the inhomogeneous Cauchy problem ⎧ ⎨ ⎩ ∂ α t uk ∂ β t u -div(|x| θ ∇u) = |u| p + ι|∇u| q + w(x), (t, x) ∈ (0, ∞) × R N , u(0, x) = u 0 (x), x ∈ R N , (1.1) where 0 ≤ θ < 2, p, q > 1, ι ≥ 0, and u 0 , w ∈ L 1 loc (R N ) with w ≡ 0. Precisely, we are concerned with the existence and nonexistence of global weak solutions to (1.1) (see Definition 2.1). The cases ι = 0 and ι > 0 are discussed separately. The nonexistence results we establish use a priori estimates for the solutions and employ appropriate contradiction arguments. The idea behind this approach is clearly presented and deeply discussed by Mitidieri and Pokhozhaev [1], who provided useful a priori estimates for solutions of the involved nonlinear equations under initial conditions, then obtained asymptotic properties of these estimates, and finally established the nonexistence proof by contradiction. Differently from the classical approach, the Mitidieri-Pokhozhaev-type approach (namely, the rescaled test-function method) does not use the maximum principle (see, for example, Pucci and Serrin [2,Ch. 3]) or other comparison results (see [2,Ch. 2]). This is as the mentioned approach does not impose any parabolicity condition on the operator or restrictive sign-conditions on the solutions. Because we also do not use any parabolicity condition, the similar arguments herein hold for the analogous elliptic problems. About the elliptic problems, we refer to the work of Papageorgiou and Scapellato [3], who prove a bifurcation-type result for a parametric nonlinear boundary value problem driven by the (p, 2)-Laplacian. In [3] the authors look for positive solutions and use a nonlinear maximum principle in their proofs. We finally mention the work of Figueiredo and Silva [4] dealing with elliptic problems defined in a half-space and involving a nonlinearity with critical growth. In [4] the authors establish the existence of positive solutions using variational methods together with Brouwer theory of degree. Now let us mention some motivations for studying problems of type (1.1). When θ = 0, ι = 0, w ≡ 0, and α, β → 1, (1.1) reduces to (1.2) When k = 0, (1.2) is just the semilinear heat equation with source term f (u) = |u| p . For k > 0, (1.2) is said to be pseudo-parabolic (see Showalter and Tin [5]). Pseudo-parabolic equations model a variety of phenomena arising in science and engineering, such as the aggregation of population [6], long waves [7], the seepage of homogeneous fluids through a fissured rock [8], and the nonstationary processes in semiconductors [9]. Cao et al. [10] investigated the large-time behavior of solutions to (1.2). Namely, like for the corresponding Cauchy problem for the semilinear heat equation (see Fujita [11]), it was shown that the Fujita critical exponent for (1.2) is equal to 1 + 2 N , which leads to the following bifurcationtype result: (i) If 1 < p ≤ 1 + 2 N and u 0 ≥ 0, u 0 ≡ 0, then any solution to (1.2) blows up in finite time; (ii) If p > 1 + 2 N and u 0 ≥ 0 is sufficiently small, then the solution to (1.2) exists globally. Indeed, the work of Fujita [11] originated a wide discussion over the link between the problem of critical exponents and the necessary conditions for the existence of solutions to evolution equations (see also [1, Part II] for more information).
On the other hand, due to the importance of fractional calculus in applications (see, e.g., [14][15][16]), a great attention was paid to the study of fractional evolution equations in many aspects. In particular, the large-time behavior of solutions to fractional space pseudo-parabolic equations was considered by many authors (see, e.g., [17][18][19]). Motivated by these contributions, in this paper, we investigate (1.1). We first consider the case ι = 0, that is, we deal with the problem For (1.5), we study the existence and nonexistence of global weak solutions and derive the Fujita critical exponent. Next, we extend our study to (1.1) with ι > 0. Our main motivation for considering this case is to study the effect of the nonlinearity |∇u| q on the critical behavior of (1.5). Namely, we show that this nonlinearity induces an interesting phenomenon of discontinuity of the Fujita critical exponent.
The rest of the paper is organized as follows. In Sect. 2, we mention in which sense solutions to (1.1) are considered and present the main results of this paper. In Sect. 3, we establish some preliminary estimates, which will be used in Sect. 4, where we prove our main results. A short Sect. 5 concludes the paper.

Main results
Let us first recall some basic notions and properties from fractional calculus. For more detail, we refer to the book of Samko et al. [16]. For f ∈ C([0, T]) with 0 < T < ∞, the Riemann-Liouville fractional integrals of order σ > 0 are given as where denotes the gamma function (see, e.g., [20]). Notice that the limit of (I σ 0 f )(t) as t goes to zero from the right is zero. So we can put (I σ 0 f )(0) = 0 to extend I σ 0 f to [0, T] by continuity. Similarly, we can extend by continuity I σ T f to [0, T] by taking (I σ T f )(T) = 0. We have the following integration-by-parts rule: (2.1) Let f ∈ C 1 ([0, T]) and σ ∈ (0, 1). The Caputo fractional derivative of order σ of f is defined by Now we present the main result obtained for (1.5). Just before, let us mention in which sense solutions to (1.5) Without loss of generality, we will suppose that k = 1.
Using the integration-by-parts rule (2.1), we define global weak solutions to (2.4) as follows.
Observe that from (2.6) we have Our main result for (2.4) is the following theorem.
Remark 2.5 From Theorem 2.2 and (2.3), and (2.6), for 0 ≤ θ < 1 and θ > 2 -N , we deduce that the Fujita critical exponent for (2.4) is given by Observe that the nonlinearity |∇u| q induces an interesting phenomenon of discontinuity of the Fujita critical exponent p * (N, θ , q) jumping from p = N θ-2+N to p = ∞, as q reaches the value N θ+N-1 from above.

Preliminary estimates
We first investigate a priori estimates of certain integral terms, which will be involved in the proofs of Theorems 2.1 and 2.2 in Sect. 4. We stress that we will consider these estimates in the application of a rescaled test-function method. This general approach to the proof of nonexistence of solutions was originally developed by Mitidieri and Pokhozhaev [1] in the case of general forms of nonlinear partial differential equations. We remark that a characteristic feature of this approach is that it requires no any comparison principle (see again [2,Ch. 2]), but uses a contradiction argument employing suitable estimates. Let us nevertheless mention that the starting point is the definition of weak solution (that is, the above method works well for solution in integral (weak) form). Here, given 0 < T < ∞ and λ 1 (λ is large enough), we consider the function Now we gather auxiliary results in the form of lemmas and discuss the validity of these estimates in R N . The first lemma gives us a useful representation formula for a Riemann-Lioville fractional integral of order ρ > 0 and its Caputo fractional derivative (recall the corresponding notions at the beginning of Sect. 2).
Hence it is sufficient to make the change of variable z = s-t T-t , and we obtain the representation Consider now a cut-off argument introducing a function ψ ∈ C ∞ ([0, ∞)) satisfying Given 0 < T < ∞, ξ > 0, and L 1, we use the cut-off function ψ ∈ C ∞ ([0, ∞)) to introduce the auxiliary function To simplify the notation, throughout this paper we denote by C positive constants independent on T and u with values changing from line to line. We start by estimating the integral of G over R N .

Lemma 3.2 We have the estimate
Proof Using the cut-off argument (3.4) for the function G given in (3.5), we have which trivially gives the desired estimate.
On this basis, we construct the next estimates.

Lemma 3.3
Let θ ≥ 0 and p > 1. Then Proof Referring to (3.5), the cut-off argument in (3.4) leads to On the other hand, an elementary calculation shows that Let us denote the inner product in R N by "·". Thus the divergent term can be written as follows: (3.8) Additionally, for the gradient term in the last line of (3.8), we have where we consider the function ν given as Taking the inner product by x on both sides of equality (3.9) and putting together the x terms in the right-hand side, we get and hence referring to (3.8), by (3.10) we obtain the estimate div |x| θ ∇ψ (3.11) To obtain the final a priori estimate, we turn to (3.6) and use (3.11) to get By assumptions L is supposed to be large enough (recall that we set L 1) and ψ ≤ 1 (see the properties of the cut-off function in (3.4)), and hence we conclude that This completes the proof.

Lemma 3.4
Let θ ≥ 0 and q > 1. Then Proof The proof follows the schema of the proof of Lemma 3.3. We use the properties of the cut-off function in (3.4) and the definition of the auxiliary function in (3.5) to obtain On the other hand, using (3.7), we obtain Again, recalling that L is supposed to be large enough and the cut-off function ψ is such that ψ ≤ 1, we have that and hence the lemma is proved.
The last result of this section is an immediate consequence of the definition of the function F given in (3.1), used together with identity (3.3) (that is, the second representation formula stated in Lemma 3.1). Therefore we omit the details of its proof.

Proofs of the main results
Proof of Theorem 2.1 As already mentioned, our arguments follow the rescaled testfunction argument in [1]. Therefore we implement a proof by contradiction. The idea is to assume that there exists a global weak solution from an appropriate class, and then using a priori asymptotic bounds for this solution, we obtain a contradiction, since there is no nontrivial global weak solution. We combine the relevant identities and inequalities established in the previous section.
(I) Suppose that u is a global weak solution to (1.5). By Definition 2.1 (focusing on (2.2)), for all 0 < T < ∞ and ϕ ∈ C 2 (Q T ) satisfying conditions (a) and (b) therein, we have the On the other hand, using Young's inequality, we obtain three inequalities related to each term in the right-hand side of (4.1). For the first term, we have Similarly, for the second term, we get and finally for the third term, we have Starting from (4.1) and using inequality (4.2) together with the inequalities (4.3) and (4.4), we deduce that where Without loss of generality, consider now the test function where F and G are given, respectively, by (3.1) and (3.5) (with λ, L 1). We can easily see that ϕ ∈ C 2 (Q T ) and it satisfies both conditions (a) and (b) of Definition 2.1. Hence the bound (4.5) holds for a function ϕ given by (4.8). Now let us estimate the integrals I i (ϕ), i = 1, 2, 3, always in the case that ϕ is given by (4.8). We have (4.9) Next, using Lemma 3.3 with θ = 0, we have that Therefore (4.9), (4.10), and (4.11) yield the estimate Proceeding in a similar way, we now estimate I 2 (ϕ). Indeed, considering (4.8), we get the identity and hence using (4.13) and (4.14) together with Lemma 3.2, we deduce that Employing a similar argument as above, by (4.8) we retrieve the identity Then by Lemma 3.3 and identity (3.14), we deduce for (4.16) the estimate Therefore it follows from (4.5), (4.12), (4.15), and (4.17) that On the other hand, by (3.5), (4.8), and (3.14) we have Notice that since w ∈ L 1 (R N ), by the dominated convergence theorem and properties of the cut-off function in (3.4), we have the asymptotic behavior Since R N w(x) dx > 0, we deduce that for sufficiently large T, Again for sufficiently large T, combining (4.19) and (4.20), it follows that On the other hand, using (4.8) and (3.2) with ρ ∈ {1α, 1 -β}, we obtain where μ T (u 0 ) = (λ + 1) We claim that Indeed, by (3.4) and (3.5), since u 0 ∈ L 1 (R N ), we have On the other hand, using (3.4) and (3.11) with θ = 0, we obtain (recall that L 1). Hence we have the estimate Therefore, passing to the limit as T → +∞ in the above inequality, the result in (4.23) is established. If we combine appropriately inequalities (4.18), (4.21), and (4.22), we get where we set Now we take ξ > 0 such that . (4.27) Notice that under the above conditions, we have A i (ξ ) < 0, i = 1, 2. Moreover, if θ ≤ 2 -N , then A 3 (ξ ) < 0 for all p > 1. Hence, passing to the limit as T → +∞ in (4.24) and using (4.23), we obtain a contradiction to the positivity condition R N w(x) dx > 0. Therefore we deduce that for all p > 1, problem (1.5) admits no global weak solution. This proves part (I)-(i) of Theorem 2.1. Next, if θ > 2 -N , then we observe that A 3 (ξ ) < 0 for all 1 < p < N θ-2+N . So, proceeding as in the previous case, we arrive again at contradiction to R N w(x) dx > 0. Thus part (I)-(ii) of Theorem 2.1 is also established. Now we focus on the second part of the theorem.
Next, we give the proof of the second main result of this manuscript (i.e., we consider (1.1) when ι > 0).
Proof of Theorem 2.2 We construct the proof following a similar strategy to the previous proof, and hence we aim to obtain a contradiction to the assumption that there exists a global weak solution to problem (2.4). We provide the precise details as follows.

Conclusions
We considered a qualitative study of sufficient and necessary conditions ensuring the existence of global weak solutions to certain classes of inhomogeneous Cauchy problems. It is noted that the presence of the parametric nonlinearity u → ι|∇u| q (namely, the case ι > 0) induces a phenomenon of discontinuity of the Fujita critical exponent in comparison with the "unperturbed" problem (namely, the case ι = 0). For other recent blow-up results and different methodologies of proofs, we refer to the works of Mohammed et al. [21] (where a fully nonlinear uniformly elliptic equation is considered using the Alexandroff-Bakelman-Pucci maximum principle) and Pan and Zhang [22] (where the mass-critical variable coefficient nonlinear Schrödinger equation is approached by the variational characterization of the ground state solutions). Finally, we cite the work of Elhindi et al. [23] dealing with a Bresse-Timoshenko model with various competing effects. The authors study both the global existence and uniqueness of solutions, employing some numerical methods (i.e., the Faedo-Galerkin approximations and multiplier techniques).