New criteria for oscillation of damped fractional partial differential equations

1.   Introduction

Differential equations of fractional order are generalizations of classical differential equations of integer order and have found varied applications in wide spread fields of science and engineering. In the last few decades, the theory of fractional differential equations and its applications have been investigated in some monographs ^[1,2,3,4,5]. In recent years, the research on the oscillatory behavior of fractional partial differential equations is a very interesting topic and some effort has been made to establish oscillation criteria for these equations which involve the Riemann-Liouville fractional partial derivative ^{[6,7,8,9,10,11,12,13,14,15]}.

However, the study of oscillatory behavior of fractional partial differential equation is still in its infancy. To develop the qualitative properties of fractional partial differential equations, it is of great interest to study the oscillatory behavior of fractional partial differential equation. In this paper, we consider the oscillatory behavior of the damped fractional partial differential equations (1.1) under boundary value conditions (1.2) and (1.3), respectively. By using Riccati type transformations and the integral averaging technique, we establish some new sufficient conditions which guarantee the oscillation of solutions of the problems $(1.1), (1.2)$ and $(1.1), (1.3)$. We also provide four examples to illustrate the relevance of the main results.

where $\alpha\in (0, 1)$ is a constant, $D^{\alpha}_{+, t} u(x, t)$ is the Riemann-Liouville fractional derivative of order $\alpha$ of $u$ with respect to $t$, $E(x, t) = \int_0^t(t-\xi)^{-\alpha}u(x, \xi)d\xi$, $\Omega$ is a bounded domain in $R^n$ with a piecewise smooth boundary $\partial\Omega$, $R_+ = (0, \infty)$, $N$ is the unit exterior normal vector to $\partial\Omega$, $\Delta$ is the Laplacianin in $R^n$ and $\beta(x, t)$ is a nonnegative continuous function on $\partial\Omega\times R_+$.

Throughout this paper, we assume that the following conditions hold:

$(C_1)$ $r\in C^1([0, \infty), R_+),$ $a, \; b, \; p\in C([0, \infty), [0, \infty)),$ $\; \tau\geq 0$ is a constant;

$(C_2)$ $q(x, t)\in C(\overline{G}, [0, \infty))$ and $q(t) = \min_{x\in\overline{\Omega}}q(x, t)$;

$(C_3)$ $f(u)\in C(R, R)$ and there exists a constant $K > 0$ such that $\frac{f(u)}{u}\geq K$ for all $u\neq 0$.

By a solution of the problem $(1.1), (1.2)$ (or $(1.1), (1.3)$), we mean a function $u(x, t)\in C^{1+\alpha}(\overline{\Omega}\times[0, \infty))$ such that $D^\alpha_{+, t}u(x, t), \; E(x, t)\in C^1(\overline{G}, R)$ and satisfies $(1.1)$ on $\overline{G}$ along with the boundary condition $(1.2)$ (or $(1.3)$).

A solution of the problem $(1.1), (1.2)$ (or $(1.1), (1.3)$) is said to be oscillatory in $G$ if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory.

In this paper, we always assume that the solutions of the problems $(1.1), (1.2)$ (or $(1.1), (1.3)$) under consideration exist globally.

The organization of the rest of this paper is as follows. In Section 2, we briefly state some basic definitions and a lemma which will be used in Section 3. In Section 3, we obtain several oscillation criteria of solutions of the problem $(1.1), (1.2)$. In Section 4, we obtain several oscillation criteria of solutions of the problem $(1.1), (1.3)$. In Section 5, we give some examples to illustrate the efficiency of our results.

2.   Preliminaries

In this section, we give the definitions of fractional derivatives and integrals and a lemma which are useful throughout this paper. There are several kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left-sided definition on the half-axis $R_+$.

Definition 2.1(^[1]). The Riemann-Liouville fractional integral of order $\alpha > 0$ for a function $y:R_+\rightarrow R$ on the half-axis $R_+$ is given by

provided the right hand side exists pointwise on $R_+$, where $\Gamma = \int^{\infty}_0e^{-t}t^{\alpha-1}dt$ is the Gamma function.

Definition 2.2(^[1]). The Riemann-Liouville fractional derivative of order $\alpha > 0$ for a function $y:R_+\rightarrow R$ on the half-axis $R_+$ is given by

provided the right hand side exists pointwise on $R_+$, where $\lceil\alpha\rceil: = \min\{z\in Z:z\geq \alpha\}$ is the ceiling function of $\alpha$.

Definition 2.3(^[3]). The Riemann-Liouville fractional derivative of order $0 < \alpha < 1$ with respect to $t$ of a function $u(x, t)$ is given by

provided the right hand side is pointwise defined on $R_+$.

Lemma 2.1(^[6]). Let

Then ${E'(t) = \Gamma(1-\alpha)D^{\alpha}_{+}y(t)}$.

3.   Oscillation of the problem $(1.1), (1.2)$

In this section, we give some sufficient conditions under which all solutions of the problem $(1.1), (1.2)$ are oscillatory.

Theorem 3.1. Suppose that the conditions $(C_1)$-$(C_3)$ hold. Furthermore, assume that for some $t_0 > 0$,

and

then every solution $u(x, t)$ of the problem $(1.1), (1.2)$ is oscillatory in $G$, where

Proof. Assume to the contrary that $u(x, t)$ is a nonoscillatory solution of the problem $(1.1), (1.2)$. Without loss of generality, we may assume that $u(x, t) > 0, \; u(x, t-\tau) > 0$ and $E(x, t) > 0$ in $\Omega\times[t_1, \infty)$ for $t_1\geq t_0$.

Integrating $(1.1)$ with respect to $x$ over the domain $\Omega$, we have

By Green's formula and the boundary condition $(1.2)$ yield

where $dS$ is the surface element on $\partial\Omega$.

By $(C_2)$ and $(C_3)$, we have

Let

then $U(t) > 0, \; t\geq t_1$. Combining $(3.3)$-$(3.6)$, we have

where

It follows from $(3.7)$ and $(3.8)$ that

Then $(r(t) D^{\alpha}_{+}U(t))v(t)$ is strictly decreasing on $[t_1, \infty)$ and thus $D^{\alpha}_{+}U(t)$ is eventually of one sign. We claim that $D^{\alpha}_{+}U(t) > 0$ on $[t_2, \infty)$, for $t_2 > t_1$ sufficiently large. In fact, if there exists a sufficiently large $t_3 > t_2$ such that $D^{\alpha}_{+}U(t_3) < 0$, then we have

Thus by Lemma 2.1 and $(3.10)$, we have

Integrating $(3.11)$ from $t_3$ to $t(t > t_3)$, we obtain

Let $t\rightarrow \infty$ in $(3.12)$, by the assumption $(3.1)$, we get $\lim_{t\rightarrow \infty}E(t) = -\infty$, which contradicts with the fact $E(t) > 0$. Hence $D^{\alpha}_{+}U(t) > 0$ for $t > t_2$.

Define

Then $W(t) > 0, \; \; t\geq t_2$. Using Lemma 2.1, from $(3.7)$ and $(3.13)$, we have

Integrating $(3.14)$ from $t_2$ to $t(t > t_2)$, we obtain

Let ${t\rightarrow \infty}$ in $(3.15)$, by the assumption $(3.2)$, we obtain a contradiction with ${W(t) > 0}$. This completes the proof of Theorem 3.1.

Theorem 3.2. Suppose that the conditions $(C_1)$-$(C_3)$ and $(3.1)$ hold, and additionally

then every solution $u(x, t)$ of the problem $(1.1), (1.2)$ is oscillatory in $G$.

Proof. We prove this theorem by contradiction. Let $u(x, t)$ be a nonoscillatory solution of the problem $(1.1), (1.2)$. One can proceed a similar proof of Theorem 3.1 to obtain $(3.7)$ and $E(t) > 0, \; D^{\alpha}_{+}U(t) > 0$ on $\Omega\times[t_2, \infty)$ for some $t_2\geq t_1\geq t_0$.

Define

Then $\tilde{W}(t) > 0$ for $t\geq t_2$. Using Lemma 2.1, from $(3.7)$ and $(3.17)$, we have

Integrating $(3.18)$ from $t_2$ to $t(t > t_2)$, we obtain

Let ${t\rightarrow \infty}$ in $(3.19)$, by the assumption $(3.16)$, we obtain a contradiction with ${\tilde{W}(t) > 0}$. This completes the proof of Theorem 3.2.

Theorem 3.3. Suppose that the conditions $(C_1)$-$(C_3)$ hold. Furthermore, assume that there exists a function $\Psi(t)\in C^1([t_0, \infty), [0, \infty))$ such that

then every solution $u(x, t)$ of the problem $(1.1), (1.2)$ is oscillatory in $G$, where

Proof. We prove by contradiction again. Suppose that $u(x, t)$ is a nonoscillatory solution of the problem $(1.1), (1.2)$. We proceed as in the similar proof of Theorem 3.1 to get $(3.7)$ and $E(t) > 0, \; D^{\alpha}_{+}U(t) > 0$ on $[t_2, \infty)$ for some $t_2\geq t_1\geq t_0$.

By Lemma 2.1, from $(3.7)$, we can get

Define

From $(3.23)$ and combining $(3.22)$, it follows that

where

Integrating $(3.24)$ from $t_2$ to $t(t > t_2)$, we obtain

From $(3.21)$, we can choose a sufficiently large $t_3$ such that

Let

then $H(t)\geq R(t) > 0, \; \; t\geq t_3$.

From $(3.26)$, we get

Divide $(3.27)$ by $R^2(t)$ and integrate it from $t_3$ to $t(t > t_3)$

Let ${t\rightarrow \infty}$ in $(3.28)$, we obtain ${\int^{\infty}_{t_3}\frac{dt}{\Psi(t)r(t)} < \infty}$, which contradicts (3.20). This completes the proof of Theorem 3.3.

In Theorem 3.3, if ${r(t)\equiv 1, \; \; \Psi(t)\equiv 1}$, then we have the following corollary.

Corollary 3.1. Suppose that the conditions $(C_1)$-$(C_3)$ hold. If

then every solution ${u(x, t)}$ of the problem $(1.1), (1.2)$ is oscillatory in $G$.

Theorem 3.4. Suppose that the conditions $(C_1)$-$(C_3)$ hold. Furthermore, assume that there exists a function ${\Psi(t)\in C^1([t_0, \infty), [0, \infty))}$, such that

then every solution ${u(x, t)}$ of the problem $(1.1), (1.2)$ is oscillatory in $G$.

Proof. Similar with the proof of Theorem 3.3, we get $(3.25)$ for $t\geq t_2$. Integrating $(3.25)$ from $t_2$ to $t(t > t_2)$ and dividing by $t$, we obtain

By $(3.30)$, we can choose a sufficiently large $t_3$ such that

Let

By using Hölder's inequality, we get

Divide $(3.32)$ by ${A^2(t)\int^t_{t_2}\Psi(s)r(s)ds}$ and integrate it from $t_3$ to $t(t > t_3)$

Let ${t\rightarrow \infty}$ in $(3.33)$ we obtain ${\int^{\infty}_{t_3}(\int^s_{t_2}\Psi(\nu)r(\nu)d\nu)^{-1}ds}$$< \infty$, which contradicts (3.29). This completes the proof of Theorem 3.4.

4.   Oscillation of the problem $(1.1), (1.3)$

The following fact in ^[16] will be used.

The first eigenvalue ${\lambda_0}$ of the Dirichlet eigenvalue problem

is positive and the corresponding eigenfunction ${\phi(x)}$ is also positive in ${\Omega}$.

Theorem 4.1. Suppose that the conditions of Theorem 3.1 hold. Then every solution $u(x, t)$ of the problem $(1.1), (1.3)$ is oscillatory in $G$.

Proof. As $u(x, t)$ is a nonoscillatory solution of the problem $(1.1), (1.3)$. Without loss of generality, we can assume that $u(x, t) > 0, \; \; u(x, t-\tau) > 0$ and $E(x, t) > 0$ in $[t_1, \infty)$ for some $t_1\geq t_0$.

By multiplying ${\phi(x)}$ on both sides of $(1.1)$ by and integrating with respect to $x$ over the domain $\Omega$, we have

Using Green's formula and the boundary condition $(1.3)$ yield

By the assumption $(C_2)$ and $(C_3)$ that

Let

then ${\tilde{U}(t) > 0, \; t\geq t_1}$. Combining $(4.1)$-$(4.4)$, we have

where

The remainder of the proof is similar to that of Theorem 3.1 and we omit it here. The proof of Theorem 4.1 is complete.

Theorem 4.2. Suppose that the conditions of Theorem 3.2 hold. Then every solution $u(x, t)$ of the problem $(1.1), (1.3)$ is oscillatory in $G$.

Theorem 4.3. Suppose that the conditions of Theorem 3.3 hold. Then every solution $u(x, t)$ of the problem $(1.1), (1.3)$ is oscillatory in $G$.

Corollary 4.1. Suppose that the conditions of Corollary 4.2 hold. Then every solution $u(x, t)$ of the problem $(1.1), (1.3)$ is oscillatory in $G$.

Theorem 4.4. Suppose that the conditions of Theorem 3.4 hold. Then every solution $u(x, t)$ of the problem $(1.1), (1.3)$ is oscillatory in $G$.

The proofs of Theorem 4.2, Theorem 4.3, Corollary 4.1 and Theorems 4.4 are similar to that of in Section 3 and hence the details are omitted.

5.   Examples

In this section, we show four examples as applications of our main results.

Example 5.1. Consider the following fractional partial differential equation

with the boundary value condition

where ${\alpha = \frac{1}{2}, \; \Omega = (0, \pi), \; n = 1, \; E(x, t) = \int_0^t(t-\xi)^{-\frac{1}{2}}u(x, \xi)d\xi}$, ${a(t) = e^t, \; b(t) = 2t}$, ${\tau = \frac{1}{3}}$, ${r(t) = \frac{1}{t^2}}$, ${p(t) = \frac{1}{t^3}}$, ${q(x, t) = x^3}$ ${+\frac{1}{t^2}}$ and $f(E(x, t)) = E(x, t)$. Hence ${q(t) = \frac{1}{t^2}}$.

Take $t_0 > 0$ and $K = 1$, we see that

and

Therefore, the conditions in Theorem 3.1 hold. Then every solution of problem $(5.1), (5.2)$ oscillates in $(0, \pi)\times R_+$.

Example 5.2. Consider the following fractional partial differential equation

with the boundary value condition

where ${\alpha = \frac{1}{2}, \; \Omega = (0, \pi), \; n = 1, \; E(x, t) = \int_0^t(t-\xi)^{-\frac{1}{2}}u(x, \xi)d\xi}$, ${a(t) = e^{-t}, \; b(t) = \frac{t}{2}, \; \tau = 1, \; r(t) = \frac{1}{t}, \; p(t) = \frac{1}{t^2}}$, ${q(x, t) = 2x^2+\frac{1}{t}}$ and $f(E(x, t)) = e^{E(x, t)}E(x, t)$. Hence ${q(t) = \frac{1}{t}}$ and ${\Gamma(1-\alpha) = \Gamma(\frac{1}{2}) = \sqrt{\pi}}$.

Take $t_0 > 0$ and $K = 1$, we have

and

Therefore, the conditions in Theorem 3.2 hold. Then every solution of problem $(5.3), (5.4)$ oscillates in $(0, \pi)\times R_+$.

Example 5.3. Consider the following fractional partial differential equation

with the boundary value condition

where ${\alpha = \frac{1}{2}, \; \Omega = (0, \pi), \; n = 1, \; E(x, t) = \int_0^t(t-\xi)^{-\frac{1}{2}}u(x, \xi)d\xi}$, ${a(t) = \frac{t}{2}, \; b(t) = 1, \; \tau = \frac{1}{2}, \; r(t) = t^2, \; p(t) = t, \; q(x, t) = \frac{2}{\sqrt{\pi}}}$ and ${f(E(x, t)) = e^{E(x, t)}E(x, t)}$. Hence ${q(t) = \frac{2}{\sqrt{\pi}}}$ and ${\Gamma(1-\alpha) = \Gamma(\frac{1}{2}) = \sqrt{\pi}}$. It is obvious that ${\lambda_0 = 1}$, ${\Phi(x) = \sin x, \; x\in \Omega}$.

Take ${t_0 > 0, \; \Psi(t) = \frac{1}{t}}$ and $K = 1$, we have

Therefore, the conditions in Theorem 4.3 hold. Then every solution of problem $(5.5), (5.6)$ oscillates in $(0, \pi)\times R_+$.

Example 5.4. Consider the following fractional partial differential equation

with the boundary value condition

where ${\alpha = \frac{1}{2}, \; \Omega = (0, \pi), \; n = 1, \; E(x, t) = \int_0^t(t-\xi)^{-\frac{1}{2}}u(x, \xi)d\xi}$, ${a(t) = 1, \; b(t) = \frac{t}{3}, \; \tau = 1, \; r(t) = e^{-t}, \; p(t) = 3e^{-t}}$, ${q(x, t) = }$ ${\frac{2e^{-t}}{\sqrt{\pi}}}$ and $f(E(x, t)) = E(x, t)$. Hence ${q(t) = \frac{2e^{-t}}{\sqrt{\pi}}}$ and ${\Gamma(1-\alpha) = \Gamma(\frac{1}{2}) = \sqrt{\pi}}$. It is obvious that ${\lambda_0 = 1}$, ${\Phi(x) = \sin x, \; x\in \Omega}$.

Take $t_0 > 0, \; \Psi(t) = e^{t}$ and $K = 1$, we see that

Therefore, the conditions in Theorem 4.4 hold. Then every solution of problem $(5.7), (5.8)$ oscillates in $(0, \pi)\times R_+$.

6.   Conclusions

In this paper, we have studied the oscillation of a class of damped fractional partial differential equations (1.1) with the Robin boundary value conditions (1.2) and the Dirichlet boundary value conditions (1.3). We have also given some new oscillation conditions by using generalized Riccati transformation method and inequality technique. We illustrated our main results by providing suitable examples. We believe that there is extensive research space on this topic.

Acknowledgments

This work is supported by the Key Scientific Research Project of Hunan Provincial Department of Education (21A0440), the Project of "Double First-Class" Applied Characteristic Discipline in Hunan Province (Xiangjiaotong[2018]469), the Project of Hunan Provincial Key Laboratory (2016TP1020), Hunan Provincial Natural Science Foundation of China (2022JJ90021) and the Discipline Special Research Projects of Hengyang Normal University (XKZX21002). The authors thank the referee very much for his valuable comments and suggestions on this paper.

Conflict of interest

All authors declare that they have no competing interests.