On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

Abstract

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

1. Introduction

Fractional derivatives have proved to be very efficient and adequate to describe many phenomena with memory and hereditary processes. These phenomena are abundant in science, engineering (viscoelasticity, control, porous media, mechanics, electrical engineering, electromagnetism, etc.) as well as in geology, rheology, finance, and biology. Unlike the classical derivatives, fractional derivatives have the ability to characterize adequately processes involving a past history. We are witnessing a huge development of fractional calculus and methods in the theory of differential equations. Indeed, after the appearance of the papers by Bagley and Torvik [1–3], researchers started to deal directly with differential equations containing fractional derivatives instead of ignoring them as it used to be the case. For analytical treatments, we may refer the reader to [4–36], and for some applications, one can consult [1–3, 8, 25, 26, 26, 27, 27–31, 33, 34, 37–49] to cite but a few.

We will consider the problem:𝒟_𝑎^𝛼,𝛽₊𝑢(𝑡)[(𝑡𝒟=𝑓𝑡,𝑢)],0<𝛼<1,0≤𝛽≤1,𝑡>𝑎>0,_{𝑎(𝛽−1)(1−𝛼)+}𝑢(𝑎)=𝑢₀≥0, (1.1) where 𝒟_𝑎^𝛼,𝛽₊𝑢 is a new type of fractional derivative we will define below and 𝑢₀ is a given constant. This new fractional derivative interpolates the Hadamard fractional derivative and its Caputo counterpart [26, 34], in the same way the Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. That is why we are naming it after Hilfer and Hadamard.

A nonexistence result for global solutions of the problem (1.1) will be proved when 𝑓[𝑡,𝑢(𝑡)]≥(log(𝑡/𝑎))^𝜇|𝑢(𝑡)|^𝑚 for some 𝑚>1 and 𝜇∈𝐑. That is we consider the Cauchy problem:𝒟_𝑎^𝛼,𝛽₊𝑢𝑡)≥𝑡(log𝑎^𝜇|𝑢(𝑡)|^𝑚𝒟,𝑡>𝑎>0,𝑚>1,𝜇∈𝐑,_{𝑎𝛾−1+}𝑢(𝑎)=𝑢₀≥0, (1.2) where 𝛾=𝛼+𝛽−𝛼𝛽 and show that no solutions can exist for all time for certain values of 𝜇 and 𝑚. Clearly, sufficient conditions for nonexistence provide necessary conditions for existence of solutions. In addition, we construct an example for which there exist solutions for some powers 𝑚 and in some appropriate space.

The existence and uniqueness of solutions for problem (1.1) has been discussed in [50] in the space 𝐶^𝛼,𝛽_{𝛿;1−𝛾,𝜇}[𝑎,𝑏] defined by𝐶^𝛼,𝛽_{𝛿;1−𝛾,𝜇}[𝑎,𝑏]=𝑦∈𝐶_1−𝛾,log[𝑎,𝑏],𝒟_𝑎^𝛼,𝛽₊𝑦∈𝐶_𝜇,log[𝑎,𝑏], (1.3) where𝐶_𝛾,log[]=(]𝑥𝑎,𝑏𝑔∶𝑎,𝑏⟶𝐑∶log𝑎^𝛾𝑔(𝑥)∈𝐶[𝑎,𝑏] (1.4) for 0≤𝜇<1 and 𝐶_0,log[𝑎,𝑏]=𝐶[𝑎,𝑏].

We also point out here that the case where 𝒟_𝑎^𝛼,𝛽₊ is the usual Riemann-Liouville fractional derivative has been studied in [26] (see also references therein). There are very few papers [26, 29] dealing with the pure Hadamard case, that is, when 𝛽=0.

The rest of the paper is divided into three sections. In Section 2, we present some definitions, notations, and lemmas which will be needed later in our proof. Section 3 is devoted to the nonexistence result and Section 4 contains an example of existence of solutions.

2. Preliminaries

In this section, we present some background material for the forthcoming analysis. For more details, see [25, 26, 33, 42, 51, 52].

Definition 2.1. The space 𝑋^𝑝_𝑐(𝑎,𝑏)(𝑐∈𝐑,1≤𝑝≤∞) consists of those real-valued Lebesgue measurable functions 𝑔 on (𝑎,𝑏) for which ‖𝑔‖_𝑋^𝑝_𝑐<∞, where 𝑔_𝑋^𝑝_𝑐=^𝑏_𝑎𝑡^𝑐𝑔(𝑡)^𝑝𝑑𝑡𝑡^1/𝑝𝑔,1≤𝑝<∞,𝑐∈𝐑,_𝑋∞𝑐=esssup_{𝑎≤𝑥≤𝑏}𝑥^𝑐𝑔(𝑥),𝑐∈𝐑. (2.1) In particular, when 𝑐=1/𝑝, we see that 𝑋^𝑝_1/𝑝(𝑎,𝑏)=𝐿_𝑝(𝑎,𝑏).

Definition 2.2. Let Ω=[𝑎,𝑏](0<𝑎<𝑏<∞) be a finite interval and 0≤𝛾<1, we introduce the weighted space 𝐶_{𝛾,𝑙𝑜𝑔}[𝑎,𝑏] of continuous functions 𝑔 on (𝑎,𝑏]: 𝐶_𝛾,log[]=(]∶𝑥𝑎,𝑏𝑔∈𝐶𝑎,𝑏log𝑎^𝛾𝑔(𝑥)∈𝐶[𝑎,𝑏]. (2.2) In the space 𝐶_𝛾,log[𝑎,𝑏], we define the norm: 𝑔_𝐶𝛾,log=𝑥log𝑎^𝛾𝑔(𝑥)_𝐶,𝑔_𝐶0,log=𝑔_∞. (2.3)

Definition 2.3. Let 𝛿=𝑥(𝑑/𝑑𝑥) be the 𝛿-derivative, for 𝑛∈𝐍, we denote by 𝐶^𝑛_𝛿,𝛾[𝑎,𝑏] (0≤𝛾<1) the Banach space of functions 𝑔 which have continuous 𝛿-derivatives on [𝑎,𝑏] up to order 𝑛−1 and have the derivative 𝛿^𝑛𝑔 of order 𝑛 on (𝑎,𝑏] such that 𝛿^𝑛𝑔∈𝐶_𝛾,log[𝑎,𝑏]: 𝐶^𝑛_𝛿,𝛾[𝑎,𝑏]=𝑔∶(𝑎,𝑏]⟶𝐑∶𝛿^𝑘𝑔∈𝐶[𝑎,𝑏],𝑘=0,…,𝑛−1,𝛿^𝑛𝑔∈𝐶_𝛾,log[𝑎,𝑏] (2.4) with the norm: 𝑔_{𝐶𝑛𝛿,𝛾}=^𝑛−1_𝑘=0𝛿^𝑘𝑔_𝐶+𝛿^𝑛𝑔_𝐶𝛾,log. (2.5) When 𝑛=0, we set 𝐶⁰_𝛿,𝛾[𝑎,𝑏]=𝐶_𝛾,log[𝑎,𝑏]. (2.6)

Definition 2.4. Let (𝑎,𝑏)(0≤𝑎<𝑏≤∞) be a finite or infinite interval of the half-axis 𝐑⁺ and let 𝛼>0. The Hadamard left-sided fractional integral 𝒥^𝛼_𝑎+𝑓 of order 𝛼>0 is defined by 𝒥^𝛼_𝑎+𝑓(𝑥)1∶=Γ(𝛼)^𝑥_𝑎𝑥log𝑡^𝛼−1𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏 (2.7) provided that the integral exists. When 𝛼=0, we set 𝒥⁰_𝑎+𝑓=𝑓. (2.8)

Definition 2.5. Let (𝑎,𝑏)(0≤𝑎<𝑏≤∞) be a finite or infinite interval of the half-axis 𝐑⁺ and let 𝛼>0. The Hadamard right-sided fractional integral 𝒥^𝛼_𝑏−𝑓 of order 𝛼>0 is defined by 𝒥^𝛼_𝑏−𝑓(𝑥)1∶=Γ(𝛼)^𝑏_𝑥𝑡log𝑥^𝛼−1𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏, (2.9) provided that the integral exists. When 𝛼=0, we set 𝒥⁰_𝑏−𝑓=𝑓. (2.10)

Definition 2.6. The left-sided Hadamard fractional derivative of order 0≤𝛼<1 on (𝑎,𝑏) is defined by 𝒟^𝛼_𝑎+𝑓(𝑥)∶=𝛿𝒥_𝑎^1−𝛼₊𝑓(𝑥), (2.11) that is, 𝒟^𝛼_𝑎+𝑓(𝑥)=𝑥𝑑1𝑑𝑥Γ(1−𝛼)^𝑥_𝑎𝑥log𝑡^−𝛼𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏. (2.12) When 𝛼=0, we set 𝒟⁰_𝑎+𝑓=𝑓. (2.13)

Definition 2.7. The right-sided Hadamard fractional derivative of order 𝛼(0≤𝛼<1) on (𝑎,𝑏) is defined by 𝒟^𝛼_𝑏−𝑓(𝑥)∶=−𝛿𝒥_𝑏^1−𝛼₋𝑓(𝑥), (2.14) that is, 𝒟^𝛼_𝑏−𝑓(𝑥)𝑥𝑑=−1𝑑𝑥Γ(1−𝛼)^𝑏_𝑥𝑡log𝑥^−𝛼𝑓(𝑡)𝑑𝑡𝑡,𝑎<𝑥<𝑏. (2.15) When 𝛼=0, we set 𝒟⁰_𝑏−𝑓=𝑓. (2.16)

Lemma 2.8. If 𝛼>0,𝛽>0 and 0<𝑎<𝑏<∞, then 𝒥^𝛼_𝑎+𝑡log𝑎^𝛽−1(𝑥)=Γ𝛽Γ𝑥𝛼+𝛽log𝑎^{𝛽+𝛼−1},𝒟^𝛼_𝑎+𝑡log𝑎^𝛽−1(𝑥)=Γ𝛽Γ𝑥𝛽−𝛼log𝑎^{𝛽−𝛼−1}. (2.17) In particular, if 𝛽=1, then the Hadamard fractional derivative of a constant is not equal to zero: 𝒟^𝛼_𝑎+1(𝑥)=1Γ()𝑥1−𝛼log𝑎^−𝛼, (2.18) when 0<𝛼<1.

Lemma 2.9. Let 0<𝑎<𝑏<∞,𝛼>0, and 0≤𝜇<1.(a) If 𝜇>𝛼>0, then 𝒥^𝛼_𝑎+ is bounded from 𝐶_𝜇,log[𝑎,𝑏] into 𝐶_{𝜇−𝛼,log}[𝑎,𝑏]. In particular, 𝒥^𝛼_𝑎+ is bounded in 𝐶_𝜇,log[𝑎,𝑏].(b)If 𝜇≤𝛼, then 𝒥^𝛼_𝑎+ is bounded from 𝐶_𝜇,log[𝑎,𝑏] into 𝐶[𝑎,𝑏]. In particular, 𝒥^𝛼_𝑎+ is bounded in 𝐶_𝜇,log[𝑎,𝑏].

This lemma justifies the following one

Lemma 2.10 (the semigroup property of the fractional integration operator 𝒥^𝛼_a⁺). Let 𝛼>0,𝛽>0, and 0≤𝜇<1. If 0<𝑎<𝑏<∞, then, for 𝑓∈𝐶_𝜇,log[𝑎,𝑏], 𝒥^𝛼_𝑎+𝒥^𝛽_𝑎+𝑓=𝒥_𝑎^𝛼+𝛽₊𝑓 (2.19) holds at any point 𝑥∈(𝑎,𝑏]. When 𝑓∈𝐶[𝑎,𝑏], this relation is valid at any point 𝑥∈[𝑎,𝑏].

Lemma 2.11. Let 0≤𝛼<1 and 0≤𝛾<1. If 𝑓∈𝐶¹_𝛾,log[𝑎,𝑏], then the fractional derivatives 𝒟^𝛼_𝑎+ and 𝒟^𝛼_𝑏− exist on (𝑎,𝑏] and [𝑎,𝑏), respectively, (𝑎>0) and can be represented in the forms: 𝒟^𝛼_𝑎+𝑓(𝑥)=𝑓(𝑎)Γ()𝑥1−𝛼log𝑎^−𝛼+1Γ(1−𝛼)^𝑥_𝑎𝑥log𝑡^−𝛼𝑓^′(𝑡)𝒟𝑑𝑡,_𝛼𝑏−𝑓𝑥)=𝑓(𝑏)(Γ()𝑏1−𝛼log𝑥^−𝛼−1Γ(1−𝛼)^𝑏_𝑥𝑡log𝑥^−𝛼𝑓^′(𝑡)𝑑𝑡, (2.20) respectively.

Lemma 2.12 (fractional integration by Parts). Let 𝛼>0 and 1≤𝑝≤∞. If 𝜑∈𝐿_𝑝(𝐑⁺) and 𝜓∈𝑋^𝑞_−1/𝑝, then ^∞₀𝜑(𝑥)𝒥^𝛼₊𝜓(𝑥)𝑑𝑥𝑥=^∞₀𝜓(𝑥)𝒥^𝛼₋𝜑(𝑥)𝑑𝑥𝑥, (2.21) where 1/𝑝+1/𝑞=1.

Definition 2.13. The fractional derivative ^𝑐𝒟^𝛼_𝑎+𝑓 of order 𝛼(0<𝛼<1) on (𝑎,𝑏) defined by ^𝑐𝒟^𝛼_𝑎+𝑓=𝒥_𝑎^1−𝛼₊𝛿𝑓, (2.22) where 𝛿=𝑥(𝑑/𝑑𝑥), is called the Hadamard-Caputo fractional derivative of order 𝛼.

Now, motivated by the Hilfer fractional derivative introduced in [41, 42], we introduce the new fractional derivative which we call Hilfer-Hadamard fractional derivative of order 0<𝛼<1 and type 0≤𝛽≤1:𝒟_𝑎^𝛼,𝛽₊𝑢(𝑡)=𝒥_𝑎^{𝛽(1−𝛼)}₊𝑡𝑑𝒥𝑑𝑡^{𝑎(1−𝛽)(1−𝛼)+}𝑢(𝑡). (2.23) The Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. This new one interpolates the Hadamard fractional derivative and its caputo counterpart. Indeed, for 𝛽=0, we find the Hadamard fractional derivative as defined in Definition 2.6 and, for 𝛽=1, we find its Caputo type counterpart (Definition 2.13).

Theorem 2.14 (Young’s inequality). If 𝑎 and 𝑏 are nonnegative real numbers and 𝑝 and 𝑞 are positive real numbers such that 1/𝑝+1/𝑞=1, then one has 𝑎𝑎𝑏≤^𝑝𝑝+𝑏^𝑞𝑞. (2.24) Equality holds if and only if 𝑎^𝑝=𝑏^𝑞.

3. Nonexistence Result

Before we state and prove our main result, we will start with the following lemma.

Lemma 3.1. If 𝛼>0 and 𝑓∈𝐶[𝑎,𝑏], then 𝒥^𝛼_𝑎+𝑓(𝑎)=lim_𝑡→𝑎𝒥^𝛼_𝑎+𝑓(𝑡)𝒥=0,_𝛼𝑏−𝑓(𝑏)=lim_𝑡→𝑏𝒥^𝛼_𝑏−𝑓(𝑡)=0. (3.1)

Proof. Since 𝑓∈𝐶[𝑎,𝑏], then on [𝑎,𝑏] we have |𝑓(𝑡)|<𝑀 for some positive constant 𝑀. Therefore, 𝒥^𝛼_𝑎+𝑓(𝑡≤1)Γ(𝛼)^𝑡_𝑎𝑡log𝑠^𝛼−1𝑓(𝑠)𝑑𝑠𝑠≤𝑀Γ(𝛼)^𝑡_𝑎𝑡log𝑠^𝛼−1𝑑𝑠𝑠≤𝑀Γ()𝑡𝛼+1log𝑎^𝛼. (3.2) As 𝛼>0, we see that 𝒥^𝛼_𝑎+𝑓(𝑎)=lim_𝑡→𝑎𝒥^𝛼_𝑎+𝑓(𝑡)=0. (3.3) In a similar manner, we prove the second part of the lemma.

The proof of the next result is based on the test function method developed by Mitidieri and Pokhozhaev in [52].

Theorem 3.2. Assume that 𝜇∈𝐑 and 𝑚<(1+𝜇)/(1−𝛾). Then, Problem (1.2) does not admit global nontrivial solutions in 𝐶^𝛾_1−𝛾,log[𝑎,𝑏], where 𝐶^𝛾_1−𝛾,log[𝑎,𝑏]=𝑦∈𝐶_1−𝛾,log[𝑎,𝑏]∶𝒟^𝛾_𝑎+𝑦∈𝐶_1−𝛾,log[𝑎,𝑏] (3.4) when 𝑢₀≥0.

Proof. Assume that a nontrivial solution exists for all time 𝑡>𝑎. Let 𝜑(𝑡)∈𝐶¹([𝑎,∞)) be a test function satisfying 𝜑(𝑡)≥0,𝜑(𝑡) is non-increasing and such that 𝜑(𝑡)∶=1,𝑎≤𝑡≤𝜃𝑇,0,𝑡≥𝑇, (3.5) for some 𝑇>𝑎 and some 𝜃 (𝜃<1) such that 𝑎<𝜃𝑇<𝑇. Multiplying the inequality in (1.2) by 𝜑(𝑡)/𝑡 and integrating over [𝑎,𝑇], we get ^𝑇_𝑎𝜑(𝑡)𝒟_𝑎^𝛼,𝛽₊𝑢(𝑡)𝑑𝑡𝑡≥^𝑇_𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡. (3.6) Observe that the integral in left-hand side exists and the one in the right-hand side exists for 𝑚<(1+𝜇)/(1−𝛾) when 𝑢∈𝐶^𝛾_1−𝛾,log[𝑎,𝑏]. Moreover, from the definition of (𝒟_𝑎^𝛼,𝛽₊𝑢)(𝑡), we can rewrite (3.6) as ^𝑇_𝑎𝜑(𝑡)𝒥_𝑎^{𝛽(1−𝛼)}₊𝑡𝑑𝒥𝑑𝑡^{𝑎1−𝛾+}𝑢(𝑡)𝑑𝑡𝑡≥^𝑇_𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡. (3.7) By virtue of Lemma 2.12 (after extending by zero outside [𝑎,𝑇]), we may deduce from (3.7) that ^𝑇_𝑎𝑑𝒥𝑑𝑡^{𝑎1−𝛾+}𝑢(𝑡)𝒥_𝑇^{𝛽(1−𝛼)}₋𝜑(𝑡)(𝑡)𝑑𝑡≥^𝑇_𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡. (3.8) Notice that Lemma 2.12 is valid in our case since ((log(𝑡/𝑎))^(1−𝛾)(𝒟^𝛾_𝑎+𝑢)∈𝐶[𝑎,𝑇] implies that |(log(𝑡/𝑎))^(1−𝛾)(𝒟^𝛾_𝑎+𝑢)(𝑡)|≤𝑀 on [𝑎,𝑇] for some positive constant 𝑀) ^𝑇_𝑎𝑡^−1/𝑝𝒟^𝛾_𝑎+𝑢(𝑡)^𝑝′𝑑𝑡𝑡≤𝑀^𝑇_𝑎𝑡^1−𝑝′𝑡log𝑎^{−𝑝′(1−𝛾)}𝑑𝑡𝑡≤𝑀^∞_𝑎𝑡^1−𝑝′𝑡log𝑎^{−𝑝′(1−𝛾)}𝑑𝑡𝑡. (3.9) Let 𝑠=(𝑝^′−1)(log(𝑡/𝑎)), then by the definition of the Gamma function, ^𝑇_𝑎𝑡^−1/𝑝𝒟^𝛾_𝑎+𝑢(𝑡)^𝑝′𝑑𝑡𝑡≤𝑀𝑎^1−𝑝′𝑝^′−1^{1−𝑝′(1−𝛾)∞0}𝑠^{−𝑝′(1−𝛾)}𝑒^−𝑠≤𝑑𝑠𝑀𝑎^1−𝑝′𝑝^′−1^{1−𝑝′(1−𝛾)}Γ1−𝑝^′1−𝛾<∞. (3.10) Hence, 𝑡(𝑑/𝑑𝑡)(𝒥_𝑎^1−𝛾₊𝑢)𝑡=(𝒟^𝛾_𝑎+𝑢)(𝑡)∈𝑋^𝑝′_−1/𝑝 (and 𝜑∈𝐿_𝑝) for some 𝑝>1/𝛾.
An integration by parts in (3.8) yields 𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝒥_𝑇^{𝛽(1−𝛼)}₋𝜑(𝑡)^𝑇_𝑡=𝑎−^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝑑𝒥𝑑𝑡^{𝑇𝛽(1−𝛼)−}𝜑(𝑡)≥𝑑𝑡_𝑇𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡, (3.11) or −𝑢₀𝒥_𝑇^{𝛽(1−𝛼)}₋𝜑𝑎⁺−^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝑑𝒥𝑑𝑡^{𝑇𝛽(1−𝛼)−}𝜑(𝑡)≥𝑑𝑡_𝑇𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡 (3.12) because (𝒥_𝑇^{𝛽(1−𝛼)}₋𝜑)(𝑇)=0 (see Lemma 3.1) and 𝒥_𝑎^1−𝛾₊𝑢𝑎⁺=𝒟_𝑎^𝛾−1₊𝑢𝑎⁺=𝑢₀. (3.13) Multiplying by 𝑡/𝑡 inside the integral in the left hand side of (3.12), we see that 𝐿∶=^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝑑−𝑡𝒥𝑑𝑡^{𝑇𝛽(1−𝛼)−}𝜑(𝑡)𝑑𝑡𝑡≥^𝑇_𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡. (3.14) It appears from Definition 2.7 that 𝐿=^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝒟_𝑇^{1−𝛽(1−𝛼)}₋𝜑(𝑡)𝑑𝑡𝑡, (3.15) and from Lemma 2.11, we see that 𝐿=^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡𝜑(𝑇))Γ𝛽()𝑇1−𝛼log𝑡^{𝛽(1−𝛼)−1}−1Γ𝛽(1−𝛼)^𝑇_𝑡𝑠log𝑡^{𝛽(1−𝛼)−1}𝜑^′(𝑠)𝑑𝑠𝑑𝑡𝑡. (3.16) Since 𝜑(𝑇)=0 and 1Γ𝛽(1−𝛼)^𝑇_𝑡𝑠log𝑡^{𝛽(1−𝛼)−1}𝜑^′(𝑠)𝑑𝑠=𝒥_𝑇^{𝛽(1−𝛼)}₋𝛿𝜑(𝑡), (3.17) the last equality becomes 𝐿=−^𝑇_𝑎𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝒥_𝑇^{𝛽(1−𝛼)}₋𝛿𝜑(𝑡)𝑑𝑡𝑡≥^𝑇_𝑎𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡. (3.18) Note that 𝛿𝜑∈𝐿_𝑝 and by the same argument as the one used at the beginning of the proof we may show that 𝒥_𝑎^1−𝛾₊𝑢∈𝑋^𝑝′_−1/𝑝 since 𝒥_𝑎^1−𝛾₊𝑢∈𝐶_1−𝛾,log[𝑎,𝑇].
Therefore, Lemma 2.12 again allows us to write 𝐿=−^𝑇_𝑎𝛿𝜑(𝑡)𝒥_𝑎^{𝛽(1−𝛼)}₊𝒥_𝑎^1−𝛾₊𝑢(𝑡)𝑑𝑡𝑡, (3.19) and by the semigroup property Lemma 2.10𝐿=−^𝑇_𝑎𝛿𝜑(𝑡)𝒥_𝑎^1−𝛼₊𝑢(𝑡)𝑑𝑡𝑡. (3.20) On the other hand, 𝛿𝜑(^𝑇_𝑎𝑡)𝒥_𝑎^1−𝛼₊𝑢(𝑡)𝑑𝑡𝑡=1Γ(1−𝛼)^𝑇_𝑎𝛿𝜑(𝑡)^𝑡_𝑎𝑡log𝑠^−𝛼𝑢(𝑠)𝑠𝑑𝑠𝑑𝑡𝑡≤1Γ(1−𝛼)^𝑇_𝑎𝛿𝜑(𝑡)^𝑡_𝑎𝑡log𝑠^−𝛼|𝑢(𝑠)|𝑠𝑑𝑠𝑑𝑡𝑡. (3.21) As 𝜑 is nonincreasing, we have 𝜑(𝑠)≥𝜑(𝑡) for all 𝑡≥𝑠 and 1/𝜑^1/𝑚(𝑠)≤1/𝜑^1/𝑚(𝑡), 𝑚>1. Also, it is clear that 𝜑^′(𝑡)=0,𝑡∈[𝑎,𝜃𝑇]. (3.22) Therefore, 1𝐿≤Γ(1−𝛼)^𝑇_𝑎𝛿𝜑(𝑡)^𝑡_𝑎𝑡log𝑠^−𝛼|𝑢(𝑠)|𝜑^1/𝑚(𝑠)𝑠𝜑^1/𝑚(𝑠)𝑑𝑠𝑑𝑡𝑡≤1Γ(1−𝛼)^𝑇_𝜃𝑇𝛿𝜑(𝑡)𝜑^1/𝑚(𝑡)^𝑡_𝑎𝑡log𝑠^−𝛼|𝑢(𝑠)|𝜑^1/𝑚(𝑠)𝑠𝑑𝑠𝑑𝑡𝑡. (3.23) Definition 2.4 allows us to write 𝐿≤^𝑇_𝜃𝑇𝛿𝜑(𝑡)𝜑^1/𝑚(𝑡)𝒥_𝑎^1−𝛼₊|𝑢|𝜑^1/𝑚(𝑡)𝑑𝑡𝑡. (3.24) By the same argument as the one used at the beginning of the proof, we may show that |𝑢(𝑡)|𝜑^1/𝑚(𝑡)∈𝑋^𝑝′_−1/𝑝(|𝑢(𝑡)|𝜑^1/𝑚(𝑡)≤|𝑢(𝑡)|). Moreover, it is easy to see that 𝛿𝜑(𝑡)/𝜑^1/𝑚(𝑡)∈𝐿_𝑝 (for, otherwise, we consider 𝜑^𝜆(𝑡) with some sufficiently large 𝜆). Thus, we can apply Lemma 2.12 to get 𝐿≤^𝑇_𝜃𝑇|𝑢(𝑡)|𝜑^1/𝑚(𝑡)𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^1/𝑚(𝑡)𝑑𝑡𝑡. (3.25) Next, we multiply by (log(𝑡/𝑎))^𝜇/𝑚.(log(𝑡/𝑎))^−𝜇/𝑚 inside the integral in the right-hand side of (3.25): 𝐿≤^𝑇_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^1/𝑚(𝑡)|𝑢(𝑡)|𝜑^1/𝑚(𝑡)log(𝑡/𝑎)^𝜇/𝑚log(𝑡/𝑎)^𝜇/𝑚𝑑𝑡𝑡. (3.26) For 𝜇≥0, we have (log(𝑡/𝑎))^−𝜇/𝑚≤(log(𝜃𝑇/𝑎))^−𝜇/𝑚 (because −𝜇/𝑚<0 and 𝑡>𝜃𝑇). It follows that 𝐿≤log𝜃𝑇𝑎^{−𝜇/𝑚𝑇}_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^1/𝑚(𝑡)𝑡log𝑎^𝜇/𝑚|𝑢(𝑡)|𝜑^1/𝑚(𝑡)𝑑𝑡𝑡. (3.27) By using the Young inequality (see Theorem 2.14), with 𝑚 and 𝑚^′ such that 1/𝑚+1/𝑚^′=1, in the right-hand side of (3.27), we find 1𝐿≤𝑚^𝑇_𝜃𝑇𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡+log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}𝑚^′𝑇_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^{1/𝑚𝑚′}(𝑡)𝑑𝑡𝑡≤1𝑚^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡+log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}𝑚^′𝑇_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^{1/𝑚𝑚′}(𝑡)𝑑𝑡𝑡. (3.28) Clearly, from (3.14) and (3.28), we see that log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}𝑚^′𝑇_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^{1/𝑚𝑚′}(𝑡)𝑑𝑡𝑡≥11−𝑚^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡, (3.29) or 1𝑚^′𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}𝑚^′𝑇_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^{1/𝑚𝑚′}(𝑡)𝑑𝑡𝑡. (3.30) Therefore, by Definition 2.5, we have ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}Γ^𝑚′(1−𝛼)^{𝑇𝜃𝑇𝑇𝑡}𝑠log𝑡^−𝛼𝛿𝜑(𝑠)𝜑^1/𝑚(𝑠)𝑑𝑠𝑠^𝑚′𝑑𝑡𝑡. (3.31) The change of variable 𝜎𝑇=𝑡 yields ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}Γ^𝑚′(1−𝛼)^{1𝜃𝑇𝜎𝑇}𝑠log𝜎𝑇^−𝛼𝜑^′(𝑠)𝜑(𝑠)^1/𝑚𝑑𝑠^𝑚′𝑑𝜎𝜎. (3.32) Another change of variable 𝑟=𝑠/𝑇 gives ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝜃𝑇/𝑎)^{−𝜇𝑚′/𝑚}Γ^𝑚′(1−𝛼)^1𝜃1𝜎𝑟log𝜎^−𝛼𝜑^′(𝑟)𝜑(𝑟)^1/𝑚𝑑𝑟^𝑚′𝑑𝜎𝜎. (3.33) We may assume that the integral term in the right-hand side of (3.33) is convergent, that is, 1Γ^𝑚′(1−𝛼)^1𝜃1𝜎𝑟ln𝜎^−𝛼𝜑^′(𝑟)𝜑(𝑟)^1/𝑚𝑑𝑟^𝑚′𝑑𝜎≤𝐶, (3.34) for some positive constant 𝐶, for otherwise we consider 𝜑^𝜆(𝑟) with some sufficiently large 𝜆. Therefore ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤𝐶log𝜃𝑇𝑎^{−𝜇𝑚′/𝑚}. (3.35) If 𝜇>0, then log𝜃𝑇𝑎^{−𝜇𝑚′/𝑚}⟶0, (3.36) as 𝑇→∞. Finally, from (3.35), we obtain lim_𝑇→∞^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡=0. (3.37) We reach a contradiction since the solution is not supposed to be trivial.
In the case 𝜇=0 we have −𝜇𝑚^′/𝑚=0 and the relation (3.35) ensures that lim_𝑇→∞^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤𝐶. (3.38) Moreover, it is clear that log𝜃𝑇𝑎^{−𝜇/𝑚𝑇}_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^1/𝑚(𝑡)𝑡log𝑎^𝜇/𝑚|𝑢(𝑡)|𝜑^1/𝑚(𝑡)𝑑𝑡𝑡≤log𝜃𝑇𝑎^{−𝜇/𝑚𝑇}_𝜃𝑇𝒥_𝑇^1−𝛼₋𝛿𝜑𝜑^{1/𝑚𝑚′}(𝑡)𝑑𝑡𝑡^{1/𝑚′𝑇}_𝜃𝑇𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡^1/𝑚. (3.39) This relation, together with (3.27) (relations (3.28) and (3.31) also are used without 𝜃), implies that ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤𝐾^𝑡_𝜃𝑇𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡^1/𝑚 (3.40) for some positive constant 𝐾, with lim_𝑇→∞^𝑇_𝜃𝑇𝑡log𝑎^𝜇|𝑢(𝑡)|^𝑚𝜑(𝑡)𝑑𝑡𝑡=0 (3.41) due to the convergence of the integral in (3.38). This is again a contradiction.
If 𝜇<0, we have (log(𝑡/𝑎))^−𝜇/𝑚≤(log(𝑇/𝑎))^−𝜇/𝑚 (because −𝜇/𝑚>0 and 𝑡<𝑇). Then, the change of variables 𝑡=(𝑇/𝑎)^𝜎 and 𝑠=(𝑇/𝑎)^𝑟 in (3.27) yields ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝑇/𝑎)^{1−𝜇𝑚′/𝑚}Γ^𝑚′(1−𝛼)^{ln𝑇/ln(𝑇/𝑎)ln𝜃𝑇/ln(𝑇/𝑎)𝜎ln𝑇/ln(𝑇/𝑎)}()ln𝑇/𝑎^𝑟(𝑇/𝑎)^𝜎−𝛼𝜑^′(𝑟)𝜑^1/𝑚(𝑟)𝑑𝑟^𝑚′𝑑𝜎, (3.42) or ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡≤log(𝑇/𝑎)^{1−𝛼𝑚′−𝜇𝑚′/𝑚}Γ^𝑚′(1−𝛼)^{ln𝑇/ln(𝑇/𝑎)ln𝜃𝑇/ln(𝑇/𝑎)𝜎ln𝑇/ln(𝑇/𝑎)}(𝑟−𝜎)^−𝛼𝜑^′(𝑟)𝜑^1/𝑚(𝑟)𝑑𝑟^𝑚′𝑑𝜎. (3.43) The expression |𝜑^′(𝑟)|/𝜑^1/𝑚(𝑟) may be assumed bounded (or else we use 𝜑^𝜆(𝑟) with a large value of 𝜆). Hence, ^𝑇_𝑎𝑡log𝑎^𝜇𝜑(𝑡)|𝑢(𝑡)|^𝑚𝑑𝑡𝑡𝑇≤𝐶log𝑎^{−𝑚′−𝜇𝑚′/𝑚} (3.44) for some positive constant 𝐶.

Although we are concerend here about nonexistence of solutions, using standard techniques, one may show the existence of local solutions of Problem (1.1) with 1<𝑚<(1+𝜇)/(1−𝛾). However, according to Theorem 3.2, such a solution cannot be continued for all time in 𝐶^𝛾_1−𝛾,log[𝑎,𝑏]. This is a phenomenon which occurs often in parabolic and hyperbolic problems with sources of polynomial type. In the absence of strong dissipations, these sources are the cause of blowup in finite time (of local solutions). For this reason, they are called blowup terms.

4. Example

For our example, we need the following lemma.

Lemma 4.1. The following result holds for the fractional derivative operator 𝒟_𝑎^𝛼,𝛽₊: 𝒟_𝑎^𝛼,𝛽₊𝑠log𝑎^𝛾−1(𝑡)=Γ𝛾Γ𝑡𝛾−𝛼log𝑎^{𝛾−𝛼−1},𝑡>𝑎;𝛾>0, (4.1) where 0<𝛼<1 and 0≤𝛽≤1.

Proof. We observe from Lemma 2.8 that 𝒥_𝑎^{(1−𝛼)(1−𝛽)}₊𝑠log𝑎^𝛾−1(𝑡)=Γ𝛾Γ()𝑡1−𝛼1−𝛽+𝛾log𝑎^{𝛾+(1−𝛼)(1−𝛽)−1}. (4.2) Therefore, 𝑡𝑑𝒥𝑑𝑡^{𝑎(1−𝛼)(1−𝛽)+}𝑠log𝑎^𝛾−1(𝑡)=𝛾+(1−𝛼)1−𝛽−1Γ𝛾Γ(𝑡1−𝛼)1−𝛽+𝛾log𝑎^{𝛾+(1−𝛼)(1−𝛽)−2}, (4.3) which, in light of the definition of 𝒟_𝑎^𝛼,𝛽₊, yields 𝒟_𝑎^𝛼,𝛽₊𝑠log𝑎^𝛾−1(𝑡)=Γ𝛾Γ()𝒥1−𝛼1−𝛽+𝛾−1^{𝑎𝛽(1−𝛼)+}𝑠log𝑎^{𝛾+(1−𝛼)(1−𝛽)−2}(𝑡). (4.4) From Lemma 2.8 again, we have 𝒟_𝑎^𝛼,𝛽₊𝑠log𝑎^𝛾−1(𝑡)=Γ𝛾Γ𝛽()()𝑡1−𝛼+𝛾+1−𝛼1−𝛽−1log𝑎^{𝛽(1−𝛼)+𝛾+(1−𝛼)(1−𝛽)−2}=Γ𝛾Γ𝑡𝛾−𝛼log𝑎^{𝛾−𝛼−1}. (4.5) The proof is complete.

Example 4.2. Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order 0<𝛼<1 and type 0≤𝛽≤1: 𝒟_𝑎^𝛼,𝛽₊𝑦𝑡)𝑡(=𝜆log𝑎^𝜇𝑦(𝑡)^𝑚(𝑡>𝑎>0;𝑚>1) (4.6) with real 𝜆, 𝜇∈𝐑⁺ (𝜆≠0). Suppose that the solution has the following form: 𝑦(𝑡)𝑡=𝑐log𝑎^𝜈. (4.7) Our aim next is to find the values of 𝑐 and 𝜈. By using Lemma 4.1 we have 𝒟_𝑎^𝛼,𝛽₊𝑐𝑠log𝑎^𝜈(𝑡)=𝑐Γ(𝜈+1)Γ()𝑡𝜈−𝛼+1log𝑎^𝜈−𝛼. (4.8) Therefore, 𝑐Γ(𝜈+1)Γ()𝑡𝜈−𝛼+1log𝑎^𝜈−𝛼𝑡=𝜆log𝑎^𝜇𝑐𝑡log𝑎^𝜈𝑚. (4.9) It can be directly shown that 𝜈=(𝛼+𝜇)/(1−𝑚) and 𝑐=[Γ((𝛼+𝜇)/(1−𝑚)+1)/𝜆Γ((𝑚𝛼+𝜇)/(1−𝑚)+1)]^1/(𝑚−1). If (𝑚𝛼+𝜇)/(1−𝑚)>−1, that is, 𝑚>(1+𝜇)/(1−𝛼), then (4.6) has the exact solution: 𝑦(𝑡)=Γ𝛼+𝜇/(1−𝑚)+1𝜆Γ𝑚𝛼+𝜇/(1−𝑚)+1^1/(𝑚−1)𝑡log𝑎^{(𝛼+𝜇)/(1−𝑚)}. (4.10) This solution satisfies the initial condition when (𝛼+𝜇)/(1−𝑚)≥𝛾−1>−1. Note that there is an overlap of the interval of existence in this example and the interval of nonexistence in the previous theorem. This may be explained by the fact that this solution is in 𝐶_1−𝛾,log[𝑎,𝑏] but not in 𝐶^𝛾_1−𝛾,log[𝑎,𝑏].

Acknowledgments

The authors wish to express their thanks to the referees for their suggestions. The authors are also very grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through the Project no. In101003.