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

Anh Tuan Nguyen; Chao Yang; Anh Tuan Nguyen; Chao Yang

doi:10.3934/era.2022071

Electronic Research Archive

2022, Volume 30, Issue 4: 1354-1373. doi: 10.3934/era.2022071

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

Anh Tuan Nguyen ^1,2,
Chao Yang ^{3
,
,}

1.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
2.
Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam
3.
College of Mathematical Sciences, Harbin Engineering University, Harbin, HLJ 150001, China

Received: 21 December 2021 Revised: 18 February 2022 Accepted: 21 February 2022 Published: 18 March 2022

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some $ L^p-L^q $-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in $ L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N) $, the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space.
- Exponential nonlinearity,
- time-space fractional diffusion,
- Caputo derivative,
- global well-posedness
Citation: Anh Tuan Nguyen, Chao Yang. On a time-space fractional diffusion equation with a semilinear source of exponential type[J]. Electronic Research Archive, 2022, 30(4): 1354-1373. doi: 10.3934/era.2022071

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Abstract

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some $ L^p-L^q $-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in $ L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N) $, the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space.

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