Initial-boundary value problems to semilinear multi-term fractional differential equations

Sergii V. Siryk; Nataliya Vasylyeva

doi:10.3934/cpaa.2023068

Article Contents

2023, Volume 22, Issue 7: 2321-2364. Doi: 10.3934/cpaa.2023068

This issue Previous Article Monotonicity of solutions for weighted fractional parabolic equations on the upper half space Next Article

Initial-boundary value problems to semilinear multi-term fractional differential equations

Sergii V. Siryk^1, and
Nataliya Vasylyeva^2,3, ,

1.
CONCEPT Lab, Istituto Italiano di Tecnologia, Via E. Melen 83, 16152, Genova, Italy
2.
Institute of Applied Mathematics and Mechanics of NASU, G.Batyuka st. 19, 84100 Sloviansk, Ukraine
3.
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

^*Corresponding author: Nataliya Vasylyeva
^*Corresponding author: Nataliya Vasylyeva

Received: March 2023

Revised: April 2023

Early access: May 15, 2023

Published online: May 15, 2023

The second author was partially supported by the Foundation of The European Federation of Academy of Sciences and Humanities (ALLEA), the Grant EFDS-FL2-08.

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Abstract

For $ \nu, \nu_i, \mu_j\in(0, 1) $, we analyze the semilinear integro-differential equation on the one-dimensional domain $ \Omega = (a, b) $ in the unknown $ u = u(x, t) $

$ {\mathbf{D}}_{t}^{\nu}(\varrho_{0}u)+\sum\limits_{i = 1}^{M}{\mathbf{D}}_{t}^{\nu_{i}}(\varrho_{i}u) -\sum\limits_{j = 1}^{N}{\mathbf{D}}_{t}^{\mu_{j}}(\gamma_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u) = g(x, t), $

where $ {\mathbf{D}}_{t}^{\nu}, {\mathbf{D}}_{t}^{\nu_{i}}, {\mathbf{D}}_{t}^{\mu_{j}} $ are Caputo fractional derivatives, $ \varrho_0 = \varrho_0(t)>0, $ $ \varrho_{i} = \varrho_{i}(t) $, $ \gamma_{j} = \gamma_{j}(t) $, $ \mathcal{L}_{k} $ are uniform elliptic operators with time-dependent smooth coefficients, $ \mathcal{K} $ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $ f $ and orders $ \nu, \nu_i, \mu_j $, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.

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Mathematics Subject Classification: Primary 35R11, 35B45, 35B65; Secondary 35Q92, 26A33, 65M22.

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Figure 1. The behavior of the function $ \omega_{1-\nu}(t) $ and the kernels $ \mathcal{N}(t;\nu, \frac{\nu}{j+1}) $ for $ j = 1, 2, 3, $ with (A) $ T^{*} = 0.01 $, (B) $ T^{*} = 0.1 $

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Figure 2. Solutions to Example 2 with $ f(x, t, u) = 0 $, $ \nu_1 = \nu/3 $, $ \mu_1 = \nu/2 $

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Figure 3. Solutions to Example 2 with $ f(x, t, u) = x t \cos(u^2) $, $ \nu_1 = \nu/3 $, $ \mu_1 = \nu/2 $

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Table 1. Values of $ \hat{\nu}^{*}_{\gamma} $ and $ \nu^{*}_{j} $, $ j = 1, 2, 3, $ for the corresponding value $ T^{*} $

$T^{*}$	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	0.11
$ \hat{\nu}^{*}_{\gamma}$	0.770	0.730	0.700	0.675	0.653	0.632	0.613	0.595	0.578	0.561	0.545
$ \nu^{*}_{1}$	0.932	0.898	0.870	0.845	0.822	0.800	0.779	0.759	0.739	0.720	0.701
$ \nu^{*}_{2}$	0.962	0.936	0.913	0.892	0.871	0.852	0.832	0.813	0.794	0.776	0.758
$ \nu^{*}_{3}$	0.972	0.951	0.931	0.912	0.893	0.875	0.856	0.838	0.821	0.803	0.785

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Table 2. Values of $ \gimel $ in Example 9.1; $ \nu_1 = \nu/3 $, $ \mu_1 = \nu/2 $

$ \nu $	$ \gimel $
$ 0.1 $	2.4443e-02	9.0291e-03	5.4896e-03
$ 0.2 $	2.4006e-02	7.8603e-03	5.0437e-03
$ 0.3 $	2.3429e-02	7.2815e-03	4.9720e-03
$ 0.4 $	2.2749e-02	6.4123e-03	4.4865e-03
$ 0.5 $	2.1996e-02	5.6072e-03	3.5239e-03
$ 0.6 $	2.1195e-02	5.4235e-03	2.4460e-03
$ 0.7 $	2.0369e-02	5.2311e-03	2.3659e-03
$ 0.8 $	1.9538e-02	5.0338e-03	2.2819e-03
$ 0.9 $	1.8722e-02	4.8379e-03	2.1973e-03
	$ K=J=10 $	$ K=J=20 $	$ K=J=30 $

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