On a time-space fractional diffusion equation with a semilinear source of exponential type

1.   Introduction

In this paper, we consider the following initial value problem for a time-space fractional diffusion equation

where $N\ge1$, $0 < \sigma\le2$, $h$ is the initial data function, and the symbol $\partial^\alpha_{t}$ stands for the Caputo derivative of fractional order $\alpha\in(0, 1)$ (Section 2). In Problem (1.1), we are mainly focus on the semilinear case in which the function $F$ satisfies the following assumptions

and

where $\nu, p > 1$ and $L$ is a positive constant. The reason why we study this source function comes from the great interest of the PDEs community with a polynomial source of the form $G_p(u) = |u|^{p-1}u$ or $G_p(u) = u^p$ and some similar forms. Many good papers about this topic have attracted our attention. Wang-Xu ^[1] and Xu-Su ^[2] used the potential well method to investigate the well-posedness of a pseudo-parabolic equation with nonlinear function $G_p$. Lian et al. ^[3] studied a Schrödinger equation with polynomial nonlinearity. They used infinite Nehari manifolds with geometric features to provide infinite sharp conditions for global existence and blowup results of solutions. A modified form of $G_p$ was considered by Chen et al. ^[4] in the Gierer–Meinhardt system. The authors applied a functional method to obtain a bound of some ratios of the solution, and then, the existence of global and blowup solutions were proved.

From strong interest of PDEs with polynomial non-linearity through the above mentioned papers and related works, we consider the following heat equation

where $u_0\in L^q({\mathbb{R}}^N), 1 < q < \infty,$ $p > 1$. Recall that (1.4) admits a scale solution

and the value $q_c = \frac{N(p-1)}{2}$ called the critical exponent plays an important role in investigating the existence and uniqueness results. Considering the case when $q = p = q_c = \frac{N}{N-2}$ in ${\mathbb{R}}^2$, the power exponent is approximated $\infty$. Therefore, it seems reasonable to replace the source function in (1.4) with the nonlinearity of the exponential type. As far as we know, the attention on nonlinear functions satisfying the assumptions (1.2) and (1.3) is derived and developed by several works in the literature ^[5,6,7,8,9]. Two groups in ^[5,8] and ^[7,9] studied, respectively, parabolic equations and the Schrödinger equation (NLS) with exponential nonlinearities. More precisely, in ^[8], Ioku proved the global-in-time existence of a mild solution to a semilinear heat equation with exponential nonlinearity under some smallness assumptions on the initial data. Meanwhile, Furioli ^[5] showed that the notions of weak and mild solutions are equivalent and investigated decay estimates and the asymptotic behavior of small-data global solutions. Nakamura and Ozawa in ^[9] provided global-in-time results for solutions in homogeneous Sobolev spaces and homogeneous Besov space to a NLS with a source function of exponential type. A source function $f(u) = (e^{4\pi|u|^2}-1-4\pi|u|^2)$ was considered for a two-dimensional NLS problem by Ibrahim et al. ^[7]. The authors showed that the solution to their problem tends to be a free Schrödinger under certain conditions. Also, we refer to some other works by Nakamura and Ozawa ^[10] and Ibrahim et al. ^[6] for wave equations with the nonlinearity of exponential growth.

During the past decades, fractional calculus has received increased attention due to its wide applications in diverse fields of science and engineering such as stochastic processes ^[11], fluid mechanics ^[12,13], chemotaxis in biology ^[14], viscoelasticity ^[15], etc. Apart from that, many interesting mathematical models and results on this topic have been done ^{[16,17,18,19,20,21,22,23,24,25]}. A strong inspiration for studying Problem (1.1) with the presence of the operator $\partial^\alpha_{t}$ comes from the fact that many physical phenomena carry the substance of diffusion processes, many studies about diffusion equations have been done ^{[26,27,28,29,30]}, and fractional calculus is very effective in modeling anomalous diffusion processes. In fact, while a diffusive particle in the usual diffusion process possesses the mean square displacement behaving like $C_1t$ for $t\to\infty,$ such behavior of a particle in an anomalous diffusion process is $C_2t^\alpha$ ^[31], $C_1, C_2$ are positive constants. Starting from the above characteristics of an anomalous diffusion process, many good works about fractional diffusion equations have been done. Because it is very difficult and lengthy to present all the related works, we would like to present only the works, which motivated us. In ^[32], a general time-fractional diffusion equation subject to the Dirichlet boundary condition was studied by Vergara and Zacher. By using energy estimates and a powerful inequality for integrodifferential operators, the authors proved sharp estimates for the decay in time of solutions. Andrade et al. ^[33] considered the following non-local initial boundary value problem associated with a time-fractional heat equation

where $\Omega\subset{\mathbb{R}}^N$ is a smooth bounded domain, $\alpha\in(0, 1),$ $\sigma\in(0, 2]$, $T_i\in{\mathbb{R}}$, $\beta_i:\Omega\to{\mathbb{R}}$, and the continuous function $f:[0, \infty)\times{\mathbb{R}}\to{\mathbb{R}}$ satisfies, $|f(t, s)|\le c(1+|s|^p)$ and

where $p > 1$ and $c$ is a positive constant. The existence and regularity of mild solutions were established with some sufficient conditions. In ^[34], Tuan et al. were concerned in a terminal value problem for a time-space fractional diffusion equation. For the problem with linear source function, regularity properties of solutions were studied. The existence, uniqueness and regularity to solution were proved in the case of nonlinear source.

The main results of this paper are providing the existence and uniqueness of mild solutions to Problem (1.1) with function $F$ satisfying (1.2) and (1.3). Corresponding to two different cases of initial data, we obtain a local-in-time solution and a global small-data solution. With usual initial data, by the Picard iteration method and some $L^p-L^q$ or $L^p-L^\Xi$ estimates of fundamental solutions involving the Mittag-Leffler function, we provide the existence and uniqueness of mild solutions to (1.1) on a reasonable time interval $(0, T]$. Apart from that, we also show that solutions are continuous from $(0, T]$ to $L^p({\mathbb{R}}^N).$ Global-in-time results are obtained by making use of the norm (3.14). From the technical point of view, we split the second term of the right-hand side of (2.1) into two parts, while the part with small-time is easy to handle, the controlling of the large time part requires small assumptions on the initial data in an Orlicz space to be achieved.

The structure of the paper is as follows. Firstly, we provided some preliminaries in section 2 including some function spaces, fractional settings and formula, linear estimates for a mild solution. The main results about the local and global well-posedness are stated in section 3.

2.   Preliminary

2.1.   Some basic setups

We first introduce some function spaces. Let $(B, \left\Vert {\cdot} \right\Vert_B)$ be a Banach space. For $T > 0$, we denote by $C\left([0, T];B\right)$ the space of all continuous functions $u$ from $[0, T]$ to $B$ and define the following space

Recall that $L^\infty\left(0, T;B\right)$ is a Banach space with respect to the norm

Let $\Xi(z): = e^{z^p}-1$. The Orlicz space ${L^\Xi({\mathbb{R}}^{N})}$ is defined as the space of all functions satisfying the following converging result for some $\kappa > 0$

The space ${L^\Xi({\mathbb{R}}^{N})}$ is a Banach space with the Luxemburg norm given as follows

Note that the space $C_0^\infty({\mathbb{R}}^N)$ is not dense in ${L^\Xi({\mathbb{R}}^{N})}$. In fact, we have the following embeddings

For more details about Orlicz spaces, we refer the reader to [37], Chapter 8] and references given there.

Next, let us provide some fractional settings. For $a, b > 0$, the Beta function $\textbf{B}$ and the Gamma function $\Gamma$ are defined respectively as follows

Let $\alpha\in(0, 1)$. By considering the following memory kernel

for a smooth enough function $u$, we can define the Caputo derivative of order $\alpha$ by

Also, for two real constants $\alpha_1$ and $\alpha_2$, we define the Mittag-Leffler function $E_{\alpha_1, \alpha_2}:\mathbb{C}\to\mathbb{C}$ in the following way

2.2.   Fundamental solutions

The fractional Laplace operator can be defined via the Fourier multiplier [5,Section 2]

where the Fourier transform is recalled as follows

and $\mathcal{F}^{-1}$ is the inverse Fourier transform. From the above definitions, for any $\alpha\in(0, 1)$ and $\sigma\in(1, 2]$, we define two functions $\mathscr{A}_1(\alpha, \sigma)(t)$ and $\mathscr{A}_2(\alpha, \sigma)(t)$ by

From [35,Section 1.3] or [14,Section 2], we see that the mild solution to Problem (1.1) satisfies the following Duhamel integral equality

It turns out that handling the operators $\mathscr{A}_1(\alpha, \sigma)(t)$ and $\mathscr{A}_2(\alpha, \sigma)(t)$ plays an important role in controlling norms of the mild solution $u$. Therefore, we provide some linear estimates of $\mathscr{A}_1(\alpha, \sigma)(t)$ and $\mathscr{A}_2(\alpha, \sigma)(t)$ in the following lemma.

Lemma 2.1. [14,Proposition 3.3] Let $r\in[1, \infty)$. Then, there exists a positive constant $C_0$ such that the following statements hold

(i) if $N > r\sigma$ and $s\in[r, \frac{Nr}{N-r\sigma})$, for any $u\in L^r({\mathbb{R}}^N)$, we have

(ii) If $N > 2r\sigma$ and $s\in[r, \frac{Nr}{N-2r\sigma})$, for any $u\in L^r({\mathbb{R}}^N)$, we have

Furthermore, (2.2) (resp. (2.3)) holds for any $s\in[r, \infty)$ if $N = r\sigma$ (resp. $N = 2r\sigma$) and $s\in[r, \infty]$ if $N < r\sigma$ (resp. $N < 2r\sigma$).

Corollary 2.2. For any $u\in {L^\Xi({\mathbb{R}}^{N})}$, we have

Proof of Corollary 2.2. First, for a positive real number $a$, we have the following observation

Therefore, the following equality holds

Next, from the expansion of the exponential function, for any $\kappa > 0$, we have

Combining this result and (2.4) yields the desired estimate. Similarly, we can find the estimate of the norm for $\mathscr{A}_2(\alpha, \sigma)(t)u$. The proof is completed.

3.   Existence and uniqueness

This section is used to present the main results of this paper including the existence and uniqueness of mild solutions satisfying Eq (2.1). We provide two different results about local well-posedness and global well-posedness according to two cases of initial data.

● For $u_0\in L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$, Problem (1.1) possesses a unique mild solution $u$ on $[0, T]$ where $T$ is small enough. In addition, this solution is also continuous on $(0, T].$

● By making some small assumptions on the initial data in $L^\Xi({\mathbb{R}}^N)$, we can prove that the solution $u$ to Problem (1.1) exists globally in time.

Theorem 3.1 (Local-in-time solution). Let $\nu, p > 1$. Suppose that $h$ belongs to $L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$. Then, we can find a reasonable number $T$ such that Problem (1.1) possesses a unique mild solution

where $\left\Vert {\cdot} \right\Vert_{L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)}: = \left\Vert {\cdot} \right\Vert_{L^p({\mathbb{R}}^N)}+\left\Vert {\cdot} \right\Vert_{ L^\infty({\mathbb{R}}^N)}$.

Proof. To begin, we consider the sequence $\{u_l\}_{l\in{\mathbb{N}}}$ as follows

We aim to prove that $\{u_l\}_{l\in{\mathbb{N}}}$ is a Cauchy sequence in $L^\infty(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N))$. Then, the completeness of this space ensures the existence of a limit function $u$ that can be shown to be the unique mild solution to Problem (1.1). To this end, the first task is to check whether $\{u_l\}_{l\in{\mathbb{N}}}$ is in $L^\infty(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N))$ or not. Indeed, we apply Lemma 2.1 to get

and

Since $h\in L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$, we easily find that

This result implies that $u_1\in L^\infty(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N))$. Before moving to the second step, we provide some nonlinear estimates of the source function. For functions $w, v\in L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$, we find that

and

We are now ready to consider the remaining elements of $\{u_l\}_{l\in{\mathbb{N}}}$. Let $R_1 = 2C_0\big\Vert {h} \big\Vert_{L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)}.$ Suppose that $u_l$ is in the open ball $B(0, R_1) \subset L^\infty(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N))$ for any $l\in{\mathbb{N}}$. From Lemma 1, the following estimate is satisfied for any $t > 0$

Apply (3.2) and the assumption (1.3), we find that

By similar arguments, we also get a same result for the $L^p$-norm as follows

where we have used (3.3) with $w = u_l$ and (1.3).

Combining the above two estimates and choosing

we obtain the following result

In view of (3.1) and (3.6), for any $l > 1$, if $u_l\in B(0, R_1)$, we obtain the estimate below

From (3.1) and (3.7), the induction method can be applied to conclude that $\{u_l\}_{l\in{\mathbb{N}}}\subset B(0, R_1).$

In addition, we can check that $\{u_l\}_{l\in{\mathbb{N}}}$ is a Cauchy sequence in $B(0, R_1)$. In fact, presume for $l\ge2$ that $u_l$ and $u_{l-1}$ are elements of $B(0, R_1)$, the techniques as in (3.4) and (3.5) enable us to find for any $t\in(0, T)$ that

and

Therefore, if we choose

the following estimate can be drawn from (3.8) and (3.9)

for any $l\ge2$. Based on this result, for any $l_2 > l_1\ge2,$ we have

It means $\{u_l\}_{l\in{\mathbb{N}}}$ is a Cauchy sequence in $B(0, R_1)$, provided that we have already shown that $\{u_l\}_{l\in{\mathbb{N}}}\subset B(0, R_1)$. By the completeness of the space $L^\infty\left(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)\right)$ and the dominated convergence theorem, there exists a unique limit function $u$ satisfying

In addition, we can also show that $u\in C\left((0, T];L^\Xi_0({\mathbb{R}}^N)\right)$. For $t, \varepsilon > 0,$ it is easy to check that

where we define

By Theorem 3.2 and Remark 1.6 in ^[36], we deduce

and

Furthermore, by using Lemma 2.1, we obtain

From the fact that $u\in B(0, R_1)\subset L^\infty\left(0, T;L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)\right)$, it follows immediately

In sum, we have

We next consider the third term on the right hand side of (3.10) as follows

where we apply Lemma 2.1. Therefore, there holds

Combining (3.10), (3.11), (3.12) and (3.13) yields the desired result. The theorem is thus proved.

Theorem 3.2. [Global small-data solution] Let $\nu > \frac{4}{3}$ and $p\ge2$. Suppose that one of the following assumptions is satisfied,

● $\nu < 2, \; \sigma < N < \frac{\sigma}{\nu-1}$, and there exists a constant $q\ge2$ satisfying

● $\nu\ge2, \; \sigma < N$, and there exists a constant $q\ge2$ satisfying

Let $\beta = \frac{\alpha\left(1-\frac{N}{\sigma q}\right)}{\nu-1}$ and $\eta = \frac{\alpha N}{\sigma \beta}$. Then, if the data of $h\in{L^\Xi({\mathbb{R}}^{N})}\cap L^\eta({\mathbb{R}}^N)\cap L^\frac{p\eta}{p+\eta}({\mathbb{R}}^N)$ is small enough, then Problem (1.1) possesses a unique mild solution in $L^\infty(0, \infty; {L^\Xi({\mathbb{R}}^{N})})$.

Remark 3.1. It's not too difficult to find a non-empty set of parameters meeting the assumptions of Theorem 3.2. Indeed, it can be pointed out some examples as follows

(i) if $\alpha = 0.4, \; \nu = 1.66, \; \sigma = 1.5$ and $p = 2$ and $N = 2$, we can choose $q = 2.1$;

(ii) if $\alpha = 0.7, \; \nu = 2.5, \; \sigma = 2$ and $p = 3$ and $N = 3$, we can choose $q = 4$.

Remark 3.2. By the embedding ${L^\Xi({\mathbb{R}}^{N})}\hookrightarrow L^\eta({\mathbb{R}}^N)$ for any $\eta\ge p$, the assumption of $h$ becomes $h\in{L^\Xi({\mathbb{R}}^{N})}\cap L^{\frac{p\eta}{p+\eta}}({\mathbb{R}}^N)$ whenever $\eta\ge p$. For example, if $\alpha = 0.7$, $\nu = 2.5, \; p = N = 3$ and $q = 4$, we have $\eta = 3.6 > p$. Therefore, we only need $h\in{L^\Xi({\mathbb{R}}^{N})}\cap L^{\frac{p\eta}{p+\eta}}({\mathbb{R}}^N)$.

Proof. We first introduce a function space for the existence of solutions as follows

where $\left\Vert {\cdot} \right\Vert_{L^\infty_\beta(0, T;{L^\Xi({\mathbb{R}}^{N})})}$ is defined by

Next, we consider the estimate of the source term. Suppose that $w, v\in {L^\Xi({\mathbb{R}}^{N})}$, we can deduce from {Taylor's} expansion of the exponential function and Hölder's inequality with ${\frac{1}{q} = \frac{1}{3q}+\frac{1}{3q}+\frac{1}{3q}}$ that

Thanks to the definition of the Luxemburg norm and the monotone convergence theorem, for any $u\in{L^\Xi({\mathbb{R}}^{N})}$ there holds

provided that

for any $q > 1$ and $z > 0.$ Therefore, we obtain the following estimate

Applying (3.16) to (3.15), we get immediately that

Using [5,Lemma 3.3], we derive

where we use the fact that for $j\in{\mathbb{N}}$, there holds $\Gamma(j+1) = j!$ and denote

where $C_1$ is a positive constant that is independent of $w, v$. Let $R_2$ be a sufficiently small constant. Suppose that $u_l$ is in an open ball $B(0, R_2) \subset L^\infty_\beta(0, T;{L^\Xi({\mathbb{R}}^{N})})$ for any $l\in{\mathbb{N}}$, we can show that $u_{l+1}\in B(0, R_2)$. In fact, on the one hand, by applying Lemma 1, we derive

Then, we use (3.17) with $w = u_{l}$ and (1.3) to find

where we presume that $R_2 < (3q)^{-\frac{1}{p}}$. On the other hand, repeat application of Lemma (2.1) with $s = p$ and $r = \frac{pq}{p+q}$ yields

where we note that if $N < \sigma q < 2\sigma q$, we deduce $p < \frac{N r}{N-2r\sigma}$. As a consequence, if $R_2 < (3r)^{-\frac{1}{p}}$, it follows from the above estimate and (3.17) that

provided that $\max\{3r(\nu-1), 3r\}\ge p$. Combining (3.18), (3.19) and the embedding $L^\infty({\mathbb{R}}^N)\cap L^p({\mathbb{R}}^N)\hookrightarrow {L^\Xi({\mathbb{R}}^{N})}$ gives

where $C_2$ is a positive constant coming from the embedding. According to the definition of the Beta function, we have

In view of (3.20) and (3.21), for any $t > 0$, the following estimate is satisfied

provided that

for any $t > 0$ and $u\in L^\infty_\beta\left(0, T;{L^\Xi({\mathbb{R}}^{N})}\right)$. Then, if $R_2$ is small enough such that

we get immediately

Next, we set

Then, for any $t\le\mathscr{T}$, (3.20) implies

provided that $u_l\in B(0, R_2)$. Since $N < \sigma q,$ we obtain

At the same time, if $t > \mathscr{T}$, we deduce

where we apply (3.22). If $R_2$ satisfies

there holds immediately

From the above results, whether $t$ is greater than $\mathscr{T}$ or $t$ is less than $\mathscr{T}$, we always get the following result

as long as $u_l\in B(0, R_2)$ and $R_2$ is small enough. For the purpose of proving $\{u_l\}_{l\in{\mathbb{N}}}$ is a subset of $B(0, R_2)$, we also need to consider the initial data $h$. On the one hand, by using Corollary 2.2, we get easily that

On the other hand, Lemma 2.1 shows that

where ${\eta} = \frac{\alpha N}{\sigma \beta} > 1$. Then, we get

Presume that the initial data is small enough, precisely,

Then, we can conclude that $u_1\in B(0, R_2).$ Hence, if $u_l\in B(0, R_2)$ for any $l\ge2$, we have $u_{l+1}\in B(0, R_2)$, provided that $R_2$ and the initial data are sufficiently small. Summarily, we have $\{u_l\}_{l\in{\mathbb{N}}}\in B(0, R_2)$.

To complete the Banach principle argument, we need also to show that $\{u_l\}_{l\in{\mathbb{N}}}$ is a Cauchy sequence in $B(0, R_2)$. Since the techniques are not too different from those in the results above, we only briefly present the main estimates. For $u_l$ and $u_{l-1}$ in $B(0, R_2)$, $l\ge2$, Lemma 2.1 yields

and

provided that $R_2 < \min\left\{(3q)^{-\frac{1}{p}}, (3r)^{-\frac{1}{p}}\right\}$. Therefore, since $L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$ embeds into ${L^\Xi({\mathbb{R}}^{N})}$, we have

It follows immediately that

Presume that $R_2$ is small enough such that

We then find that

Set

On the one hand, for any $t\le\overline{\mathscr{T}}$ and $l\ge2$, we derive (3.23) that

On the other hand, if $t > \overline{\mathscr{T}}$, the following estimate holds

as long as $R_2$ satisfies

Combining (3.25) and (3.26) yields

for any $T\in(0, \infty)$ and $l\ge2$. As the result of (3.24) and (3.27), for $u_l, u_{l-1}\in B(0, R_2)$ and $l\ge2$, we obtain

Then, by similar arguments of Theorem 3.1, we can show that $\{u_l\}_{l\in{\mathbb{N}}}$ is a Cauchy sequence. In addition, since $L^\infty_\beta\left(0, T;{L^\Xi({\mathbb{R}}^{N})}\right)$ is a complete space with the metric

there exists a unique limit function of $\{u_l\}_{l\in{\mathbb{N}}}$ in $B(0, R_2)$, which is the unique mild solution to Problem (1.1). The proof is completed.

4.   Conclusions

This paper considers a Cauchy Problem for a time-space fractional diffusion equation with exponential source term. By iteration method, a unique local mild solution is derived for initial data in $L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N)$. The existence and uniqueness are extended to be global in time when we suppose additionally that initial data in an Orlicz space are small enough. However, since the space $C_0^\infty(\mathbb{R}^N)$ is not dense in $L^\Xi(\mathbb{R}^N)$, the continuity of solutions in the term of time-variable is not considered for the global case. This will be a potential approach to improve the results of this work in the future.

Acknowledgement

The first author Anh Tuan Nguyen is funded by Van Lang University.

Conflict of interest

The authors declare there is no conflicts of interest.