Well-posedness and blow-up results for a time-space fractional diffusion-wave equation

Yaning Li; Mengjun Wang; Yaning Li; Mengjun Wang

doi:10.3934/era.2024162

Electronic Research Archive

2024, Volume 32, Issue 5: 3522-3542. doi: 10.3934/era.2024162

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Theory article

Well-posedness and blow-up results for a time-space fractional diffusion-wave equation

Yaning Li ^,,
Mengjun Wang

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received: 21 January 2024 Revised: 09 May 2024 Accepted: 14 May 2024 Published: 29 May 2024

In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.
- fractional diffusion wave equation,
- local well-posedness,
- blow up
Citation: Yaning Li, Mengjun Wang. Well-posedness and blow-up results for a time-space fractional diffusion-wave equation[J]. Electronic Research Archive, 2024, 32(5): 3522-3542. doi: 10.3934/era.2024162

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Abstract

In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.

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