Blow-up results of the positive solution for a class of degenerate parabolic equations
This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: (r(u))t=div(∣∇u∣p∇u)+f(x,t,u,∣∇u∣2),(x,t)∈D×(0,T∗),∂u∂ν+σu=0,(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),x∈D¯.\left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nab...
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2021
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oai:doaj.org-article:993919078a8748db86a416838e0413a92021-12-05T14:10:53ZBlow-up results of the positive solution for a class of degenerate parabolic equations2391-545510.1515/math-2021-0078https://doaj.org/article/993919078a8748db86a416838e0413a92021-08-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0078https://doaj.org/toc/2391-5455This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: (r(u))t=div(∣∇u∣p∇u)+f(x,t,u,∣∇u∣2),(x,t)∈D×(0,T∗),∂u∂ν+σu=0,(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),x∈D¯.\left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p>0p\gt 0, the spatial region D⊂Rn(n≥2)D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂D\partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.Dong ChenyuDing JuntangDe Gruyterarticleblow-up solutiondegenerate parabolic equationblow-up time35k9235k65MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 773-781 (2021) |
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blow-up solution degenerate parabolic equation blow-up time 35k92 35k65 Mathematics QA1-939 |
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blow-up solution degenerate parabolic equation blow-up time 35k92 35k65 Mathematics QA1-939 Dong Chenyu Ding Juntang Blow-up results of the positive solution for a class of degenerate parabolic equations |
| description |
This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: (r(u))t=div(∣∇u∣p∇u)+f(x,t,u,∣∇u∣2),(x,t)∈D×(0,T∗),∂u∂ν+σu=0,(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),x∈D¯.\left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p>0p\gt 0, the spatial region D⊂Rn(n≥2)D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂D\partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique. |
| format |
article |
| author |
Dong Chenyu Ding Juntang |
| author_facet |
Dong Chenyu Ding Juntang |
| author_sort |
Dong Chenyu |
| title |
Blow-up results of the positive solution for a class of degenerate parabolic equations |
| title_short |
Blow-up results of the positive solution for a class of degenerate parabolic equations |
| title_full |
Blow-up results of the positive solution for a class of degenerate parabolic equations |
| title_fullStr |
Blow-up results of the positive solution for a class of degenerate parabolic equations |
| title_full_unstemmed |
Blow-up results of the positive solution for a class of degenerate parabolic equations |
| title_sort |
blow-up results of the positive solution for a class of degenerate parabolic equations |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/993919078a8748db86a416838e0413a9 |
| work_keys_str_mv |
AT dongchenyu blowupresultsofthepositivesolutionforaclassofdegenerateparabolicequations AT dingjuntang blowupresultsofthepositivesolutionforaclassofdegenerateparabolicequations |
| _version_ |
1718371617814347776 |