Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source
Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-11-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-020-03083-0 |
| _version_ | 1818211903412895744 |
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| author | Abderrazak Nabti Ahmed Alsaedi Mokhtar Kirane Bashir Ahmad |
| author_facet | Abderrazak Nabti Ahmed Alsaedi Mokhtar Kirane Bashir Ahmad |
| author_sort | Abderrazak Nabti |
| collection | DOAJ |
| description | Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ for ( x , t ) ∈ R N × ( 0 , ∞ ) $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ with initial data u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ , where p , q , r > 1 $p,q,r>1$ , q ( p + r ) > q + r $q(p+r)>q+r$ , 0 < γ ≤ 2 $0<\gamma \leq 2 $ , 0 < α < 1 $0<\alpha <1$ , 0 < β ≤ 2 $0<\beta \leq 2$ , ( − Δ ) β 2 $(-\Delta )^{\frac{\beta }{2}}$ stands for the fractional Laplacian operator of order β, the weight function ν ( x ) $\nu (x)$ is positive and singular at the origin, and ∥ ⋅ ∥ q $\Vert \cdot \Vert _{q}$ is the norm of L q $L^{q}$ space. |
| first_indexed | 2024-12-12T05:39:54Z |
| format | Article |
| id | doaj.art-3438801b2df641a092ffae9e3e04a270 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-12T05:39:54Z |
| publishDate | 2020-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-3438801b2df641a092ffae9e3e04a2702022-12-22T00:35:58ZengSpringerOpenAdvances in Difference Equations1687-18472020-11-012020111010.1186/s13662-020-03083-0Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal sourceAbderrazak Nabti0Ahmed Alsaedi1Mokhtar Kirane2Bashir Ahmad3Laboratory of Mathematics and Informatics and Systems, University of Larbi TebessiNAAM, Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Sciences, King Abdulaziz UniversityNAAM, Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Sciences, King Abdulaziz UniversityNAAM, Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Sciences, King Abdulaziz UniversityAbstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ for ( x , t ) ∈ R N × ( 0 , ∞ ) $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ with initial data u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ , where p , q , r > 1 $p,q,r>1$ , q ( p + r ) > q + r $q(p+r)>q+r$ , 0 < γ ≤ 2 $0<\gamma \leq 2 $ , 0 < α < 1 $0<\alpha <1$ , 0 < β ≤ 2 $0<\beta \leq 2$ , ( − Δ ) β 2 $(-\Delta )^{\frac{\beta }{2}}$ stands for the fractional Laplacian operator of order β, the weight function ν ( x ) $\nu (x)$ is positive and singular at the origin, and ∥ ⋅ ∥ q $\Vert \cdot \Vert _{q}$ is the norm of L q $L^{q}$ space.http://link.springer.com/article/10.1186/s13662-020-03083-0Nonlocal sourceTest functionNonexistence of global solution |
| spellingShingle | Abderrazak Nabti Ahmed Alsaedi Mokhtar Kirane Bashir Ahmad Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source Advances in Difference Equations Nonlocal source Test function Nonexistence of global solution |
| title | Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source |
| title_full | Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source |
| title_fullStr | Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source |
| title_full_unstemmed | Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source |
| title_short | Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source |
| title_sort | nonexistence of global solutions of fractional diffusion equation with time space nonlocal source |
| topic | Nonlocal source Test function Nonexistence of global solution |
| url | http://link.springer.com/article/10.1186/s13662-020-03083-0 |
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