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Global Existence and Convergence of the Solution to the Nonlinear \(\psi \)-Caputo Fractional Diffusion Equation

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Abstract

This paper studies the initial-boundary value problem of a class of nonlinear time-fractional parabolic equations, where the fractional derivative used is in the sense of \(\psi \)-Caputo derivative of order \(\alpha \in (0,1)\). By using the modified \(\psi \)-Laplace transform and Fourier sine transform, the mild solution of the equation is derived. When the initial value is in an appropriate space and small enough, the global existence and uniqueness of this mild solution are proved. Furthermore, under some appropriate assumptions on the initial conditions, it is proved that when \(\alpha \rightarrow 1^-\), the mild solution of the time-fractional equation will converge to the mild solution of its classical corresponding problem. These conclusions are applicable not only to the Burgers equation but also to the Navier–Stokes equations. Finally, taking the Navier–Stokes equations as an example, the convergence is verified through numerical simulation.

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Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China

    Rui Zhu, Zhen Wang & Zhengdi Zhang

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  1. Rui Zhu
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  2. Zhen Wang
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  3. Zhengdi Zhang
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Contributions

Rui Zhu helped in writing—original draft. Zhen Wang contributed to writing—review & editing. Zhengdi Zhang validated the study.

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Correspondence to Zhen Wang.

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The authors declare there is no Conflict of interest.

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Communicated by Changpin Li.

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The work was supported by the National Natural Science Foundation of China (Grant Nos. 12101266, 12372012).

Appendices

Appendix

The Analytical Solution of Problem (1.1)

Applying the modified \(\psi \)-Laplace transform to both sides of (1.1) and utilizing Lemma 2.2, we obtain

$$\begin{aligned} s^\alpha {{\widetilde{u}}}(x,y, s) - s^{\alpha -1} u_a(x,y) - \Delta {{\widetilde{u}}}(x,y, s) = {{\widetilde{G}}}(u(x,y, s)). \end{aligned}$$
(A.1)

Further applying the Fourier sine transform to both sides gives

$$\begin{aligned} s^\alpha \widehat{{{\widetilde{u}}}}(x,y, s) - s^{\alpha -1} {{\widehat{u}}}_a(x,y) + \pi ^2 \left( \frac{m^2}{L_1^2} + \frac{n^2}{L_2^2}\right) \widehat{{{\widetilde{u}}}}(x,y,s) = \widehat{{{\widetilde{G}}}}(u(x,y,s)), \nonumber \\ \end{aligned}$$
(A.2)

which can be rearranged to solve for \(\widehat{{{\widetilde{u}}}}(x,y, s)\), i.e.,

$$\begin{aligned} \widehat{{{\widetilde{u}}}}(x,y, s)&= \frac{s^{\alpha -1}}{s^\alpha + \pi ^2 \left( \frac{m^2}{L_1^2} + \frac{n^2}{L_2^2}\right) } {{\widehat{u}}}_a(x,y) + \frac{1}{s^\alpha + \pi ^2 \left( \frac{m^2}{L_1^2} + \frac{n^2}{L_2^2}\right) } \widehat{{{\widetilde{G}}}}(u(x,y, s)). \end{aligned}$$
(A.3)

Utilizing the inverse Fourier sine transformation alongside the inverse modified \(\psi \)-Laplace transformation, we derive

$$\begin{aligned}&u(x,y, t)\nonumber \\&=\frac{4}{L_1L_2}\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }E_{\alpha , 1}\left( -\pi ^2\left( \frac{m^2}{L_1^2}+\frac{n^2}{L_2^2}\right) \left( \psi (t)-\psi (a)\right) ^\alpha \right) \nonumber \\&\quad \times \int _{0}^{L_2}\int _{0}^{L_1}u_a(x,y)\sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \textrm{d}x\textrm{d}y\cdot \sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \nonumber \\&\quad +\frac{4}{L_1L_2}\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\int _{a}^{t}\left( \psi (w)-\psi (a)\right) ^{\alpha -1}\nonumber \\&\quad \times \int _{0}^{L_2}\int _{0}^{L_1}G\Big (u\big (x,y, \psi ^{-1}\big (\psi (t)-\psi (w)+\psi (a)\big )\big ))\Big )\nonumber \\&\quad \times \sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \textrm{d}x\textrm{d}y\cdot \nonumber \\&\quad \times E_{\alpha , \alpha }\left( -\pi ^2\left( \frac{m^2}{L_1^2}+\frac{n^2}{L_2^2}\right) \left( \psi (w)-\psi (a)\right) ^\alpha \right) \psi '(w)\textrm{d}w\nonumber \\&\quad \times \sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) . \end{aligned}$$

Let \(s = \psi ^{-1}(\psi (t) - \psi (w) + \psi (a))\), then we have

$$\begin{aligned} \begin{aligned}&u(x,y,t)\\&=\frac{4}{L_1L_2}\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }E_{\alpha , 1}\left( -\pi ^2\left( \frac{m^2}{L_1^2}+\frac{n^2}{L_2^2}\right) \left( \psi (t)-\psi (a)\right) ^\alpha \right) \\&\quad \times \int _{0}^{L_2}\int _{0}^{L_1}u_a(x,y)\sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \textrm{d}x\textrm{d}y\times \sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \\&\quad +\frac{4}{L_1L_2}\int _{a}^{t}\left( \psi (t)-\psi (s)\right) ^{\alpha -1}\int _{0}^{L_2}\int _{0}^{L_1}G(u(x,y, s))\sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) \textrm{d}x\textrm{d}y\\&\quad \times E_{\alpha , \alpha }\left( -\pi ^2\left( \frac{m^2}{L_1^2}+\frac{n^2}{L_2^2}\right) \left( \psi (t)-\psi (s)\right) ^\alpha \right) \psi '(s)\textrm{d}s\cdot \sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) . \end{aligned}\nonumber \\ \end{aligned}$$
(A.4)

Define \(\lambda _{m, n}=\frac{m^2}{L_1^2}+\frac{n^2}{L_2^2},\; \phi _{m, n}=\frac{4}{L_1L_2}\sin \left( \frac{m\pi x}{L_1}\right) \sin \left( \frac{n\pi y}{L_2}\right) , \;\) and the operators

$$\begin{aligned}&{{\mathbb {S}}}_\alpha (t) v=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }E_{\alpha , 1}\left( -t^\alpha \lambda _{m, n}\right) (v, \phi _{m, n})\phi _{m, n}, \\&{{\mathbb {P}}}_\alpha (t) v=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }E_{\alpha , \alpha }\left( -t^\alpha \lambda _{m, n}\right) (v, \phi _{m, n})\phi _{m, n}. \end{aligned}$$

Thus, the analytical form of u(x, y, t) is given by

$$\begin{aligned} u(x,y, t)= & {{\mathbb {S}}}_\alpha \left( \psi (t)-\psi (a)\right) u_a+\int _{a}^{t}\left( \psi (t)-\psi (s)\right) ^{\alpha -1}\nonumber \\ & \times {{\mathbb {P}}}_\alpha (\psi (t)-\psi (s))G(u(x,y, s))\psi '(s)\textrm{d}s. \end{aligned}$$
(A.5)

Singular Gronwall Inequality

Drawing on the ideas in (Kong and Ding (2012), Theorem 2.7), we obtain the following singular Gronwall inequality.

Lemma B.1

(Singular Gronwall Inequality) Let \(\alpha , \beta , \gamma > 0\), \(\delta = \alpha + \gamma - 1 > 0\), \( \nu = \beta + \gamma - 1 > 0\), \(A \ge 0\), and let b(t) be a nonnegative, nondecreasing continuous function on the interval [a, T] such that \(0 \le b(t) \le M\) for some \(M > 0\). Additionally, assume that \(w(t) \ge 0\) and is locally integrable on [a, T]. Furthermore, suppose w(t) satisfies the inequality

$$\begin{aligned} w(t)&\le A(\psi (t) - \psi (a))^{\alpha -1} + b(t)\int _{a}^{t}(\psi (t) - \psi (s))^{\beta -1}\nonumber \\&\quad \,\, \,\times (\psi (s) - \psi (a))^{\gamma -1}w(s)\psi '(s)\textrm{d}s, \; \forall t \in [a,T]. \end{aligned}$$
(B.2)

Then, we can conclude that

$$\begin{aligned} \left\{ \begin{aligned}&w(t) \le A(\psi (t) - \psi (a))^{\alpha -1}F_{ {\nu }, \delta , \delta +\beta }\left[ \Gamma (\beta )b(t)(\psi (t) - \psi (a))^{ {\nu }}\right] , \;&A > 0, \\&w(t) = 0, \;&A = 0, \end{aligned} \right. \end{aligned}$$

where

$$\begin{aligned} F_{\rho , a, b}(z) = \sum _{k=0}^{\infty }c_kz^k, \quad c_{k+1} = \frac{\Gamma (k\rho +a)}{\Gamma (k\rho +b)}c_k, \; c_0> 0, \; b> a> 0, \; \rho > 0. \nonumber \\ \end{aligned}$$
(B.3)

Specifically, if \(\alpha = \gamma = 1\), with \(c_0 = 1\), then \(F_{ {\nu }, \delta , \delta +\beta }(z) = E_{\beta , 1}(z)\).

Proof

Define the operator \({{\mathscr {R}}}\) as

$$\begin{aligned} ({{\mathscr {R}}}w)(t) = b(t)\int _{a}^{t}(\psi (t)-\psi (s))^{\beta -1}(\psi (s)-\psi (a))^{\gamma -1}\psi '(s)w(s)\textrm{d}s. \end{aligned}$$
(B.4)

Consequently, the assumption (B.2) can be rewritten in the form

$$\begin{aligned} w(t) \le A(\psi (t)-\psi (a))^{\alpha -1} + ({{\mathscr {R}}}w)(t). \end{aligned}$$
(B.5)

From this, it is straightforward to deduce that

$$\begin{aligned} w(t) \le \sum _{k=0}^{n}({{\mathscr {R}}}^k A(\psi (t)-\psi (a))^{\alpha -1}) + ({{\mathscr {R}}}^{n+1}w)(t), \; n \in {{\mathbb {N}}}. \end{aligned}$$
(B.6)

Next, we aim to prove the inequality given by

$$\begin{aligned} ({{\mathscr {R}}}^{n}w)(t)\le \left\{ \begin{aligned}&b(t)(\Gamma (\beta )b(t))^{n-1}\prod _{i=1}^{n-1}\frac{\Gamma (i{\nu })}{\Gamma (i{\nu }+\beta )}\\&\quad \times \int _{a}^{t}(\psi (t)-\psi (s))^{n{\nu }-\gamma }\left( \psi (s)-\psi (a)\right) ^{\gamma -1}\psi '(s)w(s)\textrm{d}s,\; 0<\gamma \le 1, \\&\frac{\left( \Gamma (\beta )b(t)\right) ^n}{\Gamma (n\beta )}\left( \psi (t)-\psi (a)\right) ^{(n-1)(\gamma -1)}\\&\quad \times \int _{a}^{t}(\psi (t)-\psi (s))^{n\beta -1}(\psi (s)-\psi (a))^{\gamma -1}\psi '(s)w(s)\textrm{d}s,\; \gamma >1. \end{aligned}\right. \end{aligned}$$
(B.7)

For the case where \(0 < \gamma \le 1\), the proof will be conducted using mathematical induction. When \(n = 1\), the conclusion is evidently true. Assuming the inequality holds for n, we then obtain

$$\begin{aligned} \begin{aligned} ({{\mathscr {R}}}^{n+1}w)(t)&\le b^2(t)\left( \Gamma (\beta )b(t)\right) ^{n-1}\prod _{i=1}^{n-1}\frac{\Gamma (i\nu )}{\Gamma (i\nu +\beta )}\int _{a}^{t}(\psi (t)-\psi (s))^{\beta -1}\\&\quad \times (\psi (s)-\psi (a))^{\gamma -1}\psi '(s)\\&\quad \times \int _{a}^{s}(\psi (s)-\psi (\tau ))^{n{\nu }-\gamma }(\psi (\tau )-\psi (a))^{\gamma -1}w(\tau )\psi '(\tau )\textrm{d}\tau \textrm{d}s\\&=b(t)(\Gamma (\beta )b(t))^n\prod _{i=1}^{n}\frac{\Gamma (i{\nu })}{\Gamma (i{\nu }+\beta )}\int _{a}^{t}(\psi (\tau )-\psi (a))^{\gamma -1}w(\tau )\\&\quad \times (\psi (t)-\psi (\tau ))^{(n+1){\nu }-\gamma }\psi '(\tau )\textrm{d}\tau , \end{aligned} \end{aligned}$$

which implies that if \(0 < \gamma \le 1\), inequality (B.7) holds.

For the case where \(\gamma > 1\), we also employ mathematical induction. When \(n = 1\), inequality (B.7) holds true. Assuming the inequality is valid for n, we then derive that

$$\begin{aligned} ({{\mathscr {R}}}^{n+1}w)(t)&\le b(t)\frac{(\Gamma (\beta )b(t))^n}{\Gamma (n\beta )}\int _{a}^{t}(\psi (t)-\psi (s))^{\beta -1}(\psi (s)-\psi (a))^{n(\gamma -1)}\nonumber \\&\quad \times \int _{a}^{s}(\psi (s)-\psi (\tau ))^{n\beta -1}(\psi (\tau )-\psi (a))^{\gamma -1}w(\tau )\psi '(\tau )\textrm{d}\tau \textrm{d}s\nonumber \\&\le b(t)\frac{(\Gamma (\beta )b(t))^n}{\Gamma (n\beta )}\int _{a}^{t}(\psi (\tau )-\psi (a))^{\gamma -1}w(\tau )\psi '(\tau )\textrm{d}\tau \nonumber \\&\quad \times \int _{\tau }^{t}(\psi (s){-}\psi (\tau ))^{n\beta -1}(\psi (t){-}\psi (s))^{\beta -1}(\psi (s){-}\psi (a))^{n(\gamma -1)}\psi '(s)\textrm{d}s\nonumber \\&\le \frac{(b(t)\Gamma (\beta ))^{n+1}(\psi (t)-\psi (a))^{n(\gamma -1)}}{\Gamma ((n+1)\beta )}\nonumber \\&\quad \times \int _{a}^{t}w(\tau )(\psi (\tau )-\psi (a))^{\gamma -1}(\psi (t)-\psi (\tau ))^{(n+1)\beta -1}\psi '(\tau )\textrm{d}\tau . \end{aligned}$$
(B.8)

Thus, we have shown that inequality (B.7) holds.

Next, we will prove that \(({{\mathscr {R}}}^nw)(t) \rightarrow 0\) as \(n \rightarrow \infty \). For the case where \(0 < \gamma \le 1\), let

$$\begin{aligned} B_n = b(t)(\Gamma (\beta )b(t))^{n-1}\prod _{i=1}^{n-1}\frac{\Gamma (i {\nu })}{\Gamma (i {\nu }+\beta )}, \quad K_n(t, s) = B_n(\psi (t) - \psi (s))^{n {\nu }-\gamma }. \nonumber \\ \end{aligned}$$
(B.9)

Since \(\frac{B_{n+1}}{B_n} = \Gamma (\beta )b(t)\frac{\Gamma (n {\nu })}{\Gamma (n {\nu }+\beta )} \rightarrow 0\) as \(n \rightarrow \infty \) (Lemma 3.1 is used), it implies that \(K_n(t, s) \rightarrow 0\) as \(n \rightarrow \infty \). Hence, we obtain \(({{\mathscr {R}}}^nw)(t) \rightarrow 0\) as \(n \rightarrow \infty \). Similarly, when \(\gamma > 1\), we have the same results. Then, when \(A = 0\), letting \(n \rightarrow \infty \), we obtain \(w(t) = 0\).

Finally, we claim that

$$\begin{aligned} ({{\mathscr {R}}}^kA(\psi (t)-\psi (a))^{\alpha -1})(t) \le A(\Gamma (\beta )b(t))^kc_k(\psi (t)-\psi (a))^{\alpha -1+k {\nu }}, \end{aligned}$$
(B.10)

where \(c_0 = 1, \; c_k = \prod _{i=0}^{k-1}\left( \frac{\Gamma (i {\nu }+\delta )}{\Gamma (i {\nu }+\delta +\beta )}\right) ,\; k \in {{\mathbb {N}}}^+\).

When \(k=0\), it is obvious to prove that (B.10) holds. Suppose that (B.10) holds for k, then consider the estimate

$$\begin{aligned} \begin{aligned}&({{\mathscr {R}}}^{k+1}A(\psi (t)-\psi (a))^{\alpha -1})(t) \\&= b(t)\int _{a}^{t}(\psi (t){-}\psi (s))^{\beta -1}(\psi (s){-}\psi (a))^{\gamma -1}({{\mathscr {R}}}^kA(\psi (s){-}\psi (a))^{\alpha -1})(s)\psi '(s)\textrm{d}s \\&\le Ab(t)(\Gamma (\beta )b(t))^k(\psi (t)-\psi (a))^{\alpha +\beta +\gamma +k {\nu }-2}\frac{\Gamma (\beta )\Gamma (\alpha +k {\nu }+\gamma -1)}{\Gamma (\alpha +\beta +\gamma +k {\nu }-1)}c_k \\&= A(\Gamma (\beta )b(t))^{k+1}(\psi (t)-\psi (a))^{\alpha +(k+1) {\nu }-1}c_{k+1}, \end{aligned} \end{aligned}$$
(B.11)

which implies (B.10) also holds for \(k+1\). According to mathematical induction, (B.10) holds for any natural number k. Hence, we obtain

$$\begin{aligned} \begin{aligned} w(t)&\le \sum _{k=0}^{\infty }A(\Gamma (\beta )b(t))^kc_k(\psi (t)-\psi (a))^{\alpha -1+k {\nu }} \\&\le A(\psi (t)-\psi (a))^{\alpha -1}F_{ {\nu }, \delta , \delta +\beta }\big (\Gamma (\beta )b(t)(\psi (t)-\psi (a))^{ {\nu }}\big ). \end{aligned} \end{aligned}$$
(B.12)

The proof is completed. \(\square \)

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Zhu, R., Wang, Z. & Zhang, Z. Global Existence and Convergence of the Solution to the Nonlinear \(\psi \)-Caputo Fractional Diffusion Equation. J Nonlinear Sci 35, 36 (2025). https://doi.org/10.1007/s00332-025-10129-8

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