Remarks on Blow-Up Phenomena in $p$-Laplacian Heat Equation with Inhomogeneous Nonlinearity

Majdoub, Eadah Ahmad Alzahrani Mohamed

doi:10.4208/jpde.v34.n1.3

JPDE

2021

DOI: 10.4208/jpde.v34.n1.3

|View full text |Cite

Remarks on Blow-Up Phenomena in $p$-Laplacian Heat Equation with Inhomogeneous Nonlinearity

Eadah Ahmad Alzahrani Mohamed Majdoub¹

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

2023

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

New blow‐up conditions to p‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

Chung

Hwang

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we study blow‐up phenomena of the following p‐Laplace type nonlinear parabolic equations ut=∇·ρ(|∇u|p)|∇u|p−2∇u+f(x,t,u),inΩ×(0,t∗), under nonlinear mixed boundary conditions ρ(|∇u|p)|∇u|p−2∂u∂n+θ(z)ρ(|u|p)|u|p−2u=h(z,t,u),onΓ1×(0,t∗), and u=0 on Γ2 × (0, t∗) such that normalΓ1∪normalΓ2=∂normalΩ, where f and h are real‐valued C1‐functions. To discuss blow‐up solutions, we introduce new conditions: For each x ∈ Ω, z ∈ ∂Ω, t > 0, u > 0, and v > 0, false(Dp0.1em1false)0.1em:0.1emαFfalse(x,t,ufalse)≤uffalse(x,t,ufalse)+β1up+γ1,3.7emαHfalse(z,t,ufalse)≤uhfalse(z,t,ufalse)+β2up+γ2,false(Dp0.1em2false)0.1em:0.1emδvρfalse(vfalse)≤Pfalse(vfalse), for some constants α, β1, β2, γ1, γ2, and δ satisfying α>2,δ>0,β1+λR+1λSβ2≤αδp−1ρmλR,and0≤β2≤αδp−1ρmλS, where ρm:=infw>0ρfalse(wfalse), Pfalse(vfalse)=∫0vρfalse(wfalse)dw, Ffalse(x,t,ufalse)=∫0uffalse(x,t,wfalse)dw, and Hfalse(x,t,ufalse)=∫0uhfalse(x,t,wfalse)dw. Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p‐Laplace operator, respectively.

show abstract

New blow‐up conditions to p‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

Chung

Hwang

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

Mathematical analysis and numerical simulation of a strongly nonlinear singular model

Maryem,

Laila,

Nour Eddine

et al. 2023

AUCMCS

View full text Add to dashboard Cite

In this paper, we are interested in the one-dimensional singular optimization problem with constraints: \begin{equation*} \text{Min} \left\lbrace \mathcal{J}(v) = \frac{1}{p} \displaystyle \int_{-1}^{1} \left| v_{x} \right|^p+ \ \frac{1}{\gamma-1} \displaystyle \int_{-1}^{1} v^{1-\gamma},\\ \\ v(\pm 1)=0 \ \text{and} \ v(0)=d \right\rbrace, \end{equation*} where $1p\infty$, $1 \gamma \frac{2p-1}{p-1}$ and $d>0$. In the first part of the paper, we show the existence of a critical value $d^{*}>0$ such that if $d \leq d^{*}$, $\mathcal{J}$ admits a minimum in a carefully chosen closed convex set of $W^{1,p}_{0}(-1,1)$. The second part of the paper is dedicated to numerical simulations. We elaborate a numerical algorithm that transforms our constrained optimization problem into the solution of a system of ordinary differential equations. Illustrative examples are given to verify the efficiency and accuracy of the proposed numerical method to test the relevance of the proposed approach. We point out that the numerical results obtained are in good agreement with the physical phenomenon of pleated graphene in the particular case p=4 and $\gamma=9/5$ [12].

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Remarks on Blow-Up Phenomena in $p$-Laplacian Heat Equation with Inhomogeneous Nonlinearity

Cited by 2 publications

References 15 publications

New blow‐up conditions to p‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

New blow‐up conditions to p‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

Mathematical analysis and numerical simulation of a strongly nonlinear singular model

Contact Info

Product

Resources

About