Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Goldstein, Gisèle Ruiz; Goldstein, Jerome A.; Kömbe, Ismail; Tellioğlu, Reyhan

doi:10.1007/s10231-022-01226-6

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…It is worth mentioning here that, along with our model problem () unifying both problems () and (), our new Theorem 1 recovers the results in Goldstein et al 26,27 …”

Section: Introductionsupporting

confidence: 81%

“…We utilize the family of test functions

{\phi}_{\epsilon }

used in Goldstein et al 18,27 to prove Corollaries 1–3.…”

Section: Nonexistence Of Positive Solutionsmentioning

confidence: 99%

“…Finally, we close this section by highlighting two additional studies that motivate us in this topic. Among the recent developments with regard to the Robin boundary condition are our joint articles 26,27 with Goldstein and Goldstein, in which we studied the nonlinear parabolic Robin problems,…”

Section: Introductionmentioning

confidence: 99%

\left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;amp;amp;#x0003D;\Delta \left({u}&amp;amp;amp;#x0005E;m\right)&amp;amp;amp;#x0002B;V(x){u}&amp;amp;amp;#x0005E;m&amp;amp;amp;#x0002B;\lambda {u}&amp;amp;amp;#x0005E;q\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;amp;amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega, \\ {}\frac{\partial u}{\partial \nu }&amp;amp;amp;#x0003D;\beta (x)u\kern0.30em &amp;amp;amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right),\end{array}\right.

and

({, \begin{matrix} array \frac{\partial u}{\partial t} = Δ_{p} u + V (x) u^{p - 1} + λ u^{q} & array in Ω \times (0, T), \\ array u (x, 0) = u_{0} (x) \geq 0 & array in \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

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Doubly nonlinear parabolic equations with Robin boundary conditions

Kömbe

Balekoğlu

2022

Math Methods in App Sciences

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In this article, we consider the nonlinear problem {array∂u∂t=Au+V(x)um+p−2+λuqarrayinΩ×(0,T),arrayu(x,0)=u0(x)≥0arrayinΩ,array|∇u|p−2∂u∂ν=β(x)up−1arrayon∂Ω×(0,T).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;#x0003D;\mathbf{A}u&amp;#x0002B;V(x){u}&amp;#x0005E;{m&amp;#x0002B;p-2}&amp;#x0002B;\lambda {u}&amp;#x0005E;q\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega, \\ {}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\frac{\partial u}{\partial \nu }&amp;#x0003D;\beta (x){u}&amp;#x0005E;{p-1}\kern0.30em &amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$ where boldAu=divfalse(mum−1false|∇ufalse|p−2∇ufalse)$$ \mathbf{A}u&amp;#x0003D;\operatorname{div}\left(m{u}&amp;#x0005E;{m-1}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, normalΩ⊂ℝN$$ \Omega \subset {\mathbb{R}}&amp;#x0005E;N $$ is a bounded domain with smooth boundary, m>0,0.1em10,0.1emλ∈ℝ,0.1emq>0$$ m&gt;0,1&lt;p&lt;N,V\in {L}_{loc}&amp;#x0005E;1\left(\Omega \right),m&amp;#x0002B;p-2&gt;0,\lambda \in \mathbb{R},q&gt;0 $$ and β∈Lloc1false(∂normalΩfalse)$$ \beta \in {L}_{loc}&amp;#x0005E;1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions Vfalse(xfalse),0.1emβfalse(xfalse)$$ V(x),\beta (x) $$, and the exponents m+p$$ m&amp;#x0002B;p $$ and q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials Vfalse(xfalse)$$ V(x) $$ are taken into account to demonstrate applications of our main result.

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“…It is worth mentioning here that, along with our model problem () unifying both problems () and (), our new Theorem 1 recovers the results in Goldstein et al 26,27 …”

Section: Introductionsupporting

confidence: 81%

“…We utilize the family of test functions

{\phi}_{\epsilon }

used in Goldstein et al 18,27 to prove Corollaries 1–3.…”

Section: Nonexistence Of Positive Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;amp;amp;#x0003D;\Delta \left({u}&amp;amp;amp;#x0005E;m\right)&amp;amp;amp;#x0002B;V(x){u}&amp;amp;amp;#x0005E;m&amp;amp;amp;#x0002B;\lambda {u}&amp;amp;amp;#x0005E;q\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;amp;amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega, \\ {}\frac{\partial u}{\partial \nu }&amp;amp;amp;#x0003D;\beta (x)u\kern0.30em &amp;amp;amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right),\end{array}\right.

and

({, \begin{matrix} array \frac{\partial u}{\partial t} = Δ_{p} u + V (x) u^{p - 1} + λ u^{q} & array in Ω \times (0, T), \\ array u (x, 0) = u_{0} (x) \geq 0 & array in \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

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Doubly nonlinear parabolic equations with Robin boundary conditions

Kömbe

Balekoğlu

2022

Math Methods in App Sciences

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“…Moreover, in this reference, the authors used the comparison principle for the heat equation, which cannot be applied for our problem (1.1) in the case . For other contributions related to the study of parabolic equations with Hardy–Leray potential in bounded domains, see for example [2–4, 16, 33] and the references therein. To the best of our knowledge, the study of sign-changing solutions to evolution equations or inequalities involving Hardy–Leray potential in bounded domains has not been previously considered in the literature.…”

Section: Introductionmentioning

confidence: 99%

Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain

Chen¹,

Jleli²,

Samet³

2023

Proceedings of the Edinburgh Mathematical Society

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We consider a class of nonlinear higher-order evolution inequalities posed in $(0,\infty)\times B_1\backslash\{0\}$ , subject to inhomogeneous Dirichlet-type boundary conditions, where B1 is the unit ball in $\mathbb{R}^N$ . The considered class involves differential operators of the form \begin{equation*} \mathcal{L}_{\mu_1,\mu_2}=-\Delta +\frac{\mu_1}{|x|^2}x\cdot \nabla +\frac{\mu_2}{|x|^2},\qquad x\in \mathbb{R}^N\backslash\{0\}, \end{equation*} where $\mu_1\in \mathbb{R}$ and $\mu_2\geq -\left(\frac{\mu_1-N+2}{2}\right)^2$ . Optimal criteria for the nonexistence of weak solutions are established. Our study yields naturally optimal nonexistence results for the corresponding class of elliptic inequalities. Notice that no restriction on the sign of solutions is imposed.

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Improvement of the Hardy Inequality and Legendre Polynomials

Zographopoulos

2023

Springer Optimization and Its Applications

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Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Cited by 6 publications

References 20 publications

Doubly nonlinear parabolic equations with Robin boundary conditions

Doubly nonlinear parabolic equations with Robin boundary conditions

Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain

Improvement of the Hardy Inequality and Legendre Polynomials

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