In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN)...In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN).To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions,we establish the propagation estimate and the growth estimate for the solutions.It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.展开更多
In this survey,we present a literature review on the study of travel-ing waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy.The main results on wave...In this survey,we present a literature review on the study of travel-ing waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy.The main results on wave exis-tence and stability are presented for the typical degenerate equations,including porous medium equations,flux limited diffusion equations,delayed degener-ate diffusion equations,and other strong degenerate diffusion equations.展开更多
In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite p...In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite propagation property and the L^(1)−L^(∞)smoothing effect,we find that the complicated asymptotic behavior of the rescaled solutions t^(μ/2)u(t^(β)⋅,t)for 0<μ<2 N/(N[m(p−1)−1]+p)andβ>(2−μ[m(p−1)−1])/(2 p)can take place.展开更多
In this paper,the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection-div(|x|^(α)|■u|^(p-2)■u)+λ^(1)/|x|β|■u|^(p-2)■u·x=...In this paper,the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection-div(|x|^(α)|■u|^(p-2)■u)+λ^(1)/|x|β|■u|^(p-2)■u·x=|x|^(γ)g(|x|)in B{0}subject to nonlinear Robin boundary value condition|x|^(α)|■u|^(p-2)■u·n=A-ρu on■B,whereλ>0,B■RN(N≥2)is the unit ball centered at the origin,α>0,p>1,β∈R,γ>-N,g∈C(0,1])with g(0)>0,A∈R,ρ>0 and n is the unit outward normal.The same problem with diffusion promoted by convection,namely λ≤0,has already been discussed by the last two authors(Song-Yin(2012)),where the existence,nonexistence and classification of singularities for solutions are presented.Completely different from[Song,H.J.and Yin,J.X.,Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection,Math.Meth.Appl.Sci.,35,2012,1089-1100],in the present caseλ>0,namely the diffusion is suppressed by the convection,non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point.At the same time,the singular solutions may exist only if the diffusion dominates the convection.展开更多
基金This research was supported in part by NSFC(11771156 and 11371153)NSF of CQ(cstc2019jcyj-msxmX0381)+1 种基金Chongqing Municipal Key Laboratory of Institutions of Higher Education(Grant No.[2017]3)Research project of Chongqing Three Gorges University(17ZP13).
文摘In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN).To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions,we establish the propagation estimate and the growth estimate for the solutions.It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.
基金The research of T.Xu was supported by Guangdong Basic and Applied Basic Research Foundation Grant No.2021A1515110951by China Postdoctoral Science Foundation Grant No.2021M691070+2 种基金by Guangzhou Basic and Applied Basic Research Foundation,NSFC Grant Nos.11971179 and 11871230NSF of Guangdong Grant No.2019A1515010993J.Yin was supported by NSFC Grant Nos.12171166 and 12126204.
文摘In this survey,we present a literature review on the study of travel-ing waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy.The main results on wave exis-tence and stability are presented for the typical degenerate equations,including porous medium equations,flux limited diffusion equations,delayed degener-ate diffusion equations,and other strong degenerate diffusion equations.
基金This research was supported by the NSFC(Grant No.12171166)by the NSF of CQ(Grant No.cstc2019jcyj-msxmX0381)by the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant Nos.KJZD-M202001201,KJZD-M202201202).
文摘In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite propagation property and the L^(1)−L^(∞)smoothing effect,we find that the complicated asymptotic behavior of the rescaled solutions t^(μ/2)u(t^(β)⋅,t)for 0<μ<2 N/(N[m(p−1)−1]+p)andβ>(2−μ[m(p−1)−1])/(2 p)can take place.
基金supported by the National Natural Science Foundation of China(Nos.12171166,11601200,11771156,11861038,12161045,12026220)。
文摘In this paper,the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection-div(|x|^(α)|■u|^(p-2)■u)+λ^(1)/|x|β|■u|^(p-2)■u·x=|x|^(γ)g(|x|)in B{0}subject to nonlinear Robin boundary value condition|x|^(α)|■u|^(p-2)■u·n=A-ρu on■B,whereλ>0,B■RN(N≥2)is the unit ball centered at the origin,α>0,p>1,β∈R,γ>-N,g∈C(0,1])with g(0)>0,A∈R,ρ>0 and n is the unit outward normal.The same problem with diffusion promoted by convection,namely λ≤0,has already been discussed by the last two authors(Song-Yin(2012)),where the existence,nonexistence and classification of singularities for solutions are presented.Completely different from[Song,H.J.and Yin,J.X.,Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection,Math.Meth.Appl.Sci.,35,2012,1089-1100],in the present caseλ>0,namely the diffusion is suppressed by the convection,non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point.At the same time,the singular solutions may exist only if the diffusion dominates the convection.
