scholarly journals Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 394
Author(s):  
Awatif Alqahtani ◽  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Weak Solutions ◽  
Neumann Boundary ◽  
Large Time Behavior ◽  
Open Unit ◽  
Normal Vector ◽  
Time Behavior ◽  
Global Weak Solutions ◽  
Open Unit Ball ◽  
Test Function ◽  
Outward Unit

We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = κ | u ( t , x ) | p , ( t , x ) ∈ ( 0 , ∞ ) × D c under the nonhomegeneous Neumann boundary condition ∂ u ∂ ν ( t , x ) = λ ( x ) , ( t , x ) ∈ ( 0 , ∞ ) × ∂ D , where L : = i ∂ t + Δ is the Schrödinger operator, D = B ( 0 , 1 ) is the open unit ball in R N , N ≥ 2 , D c = R N ∖ D , p > 1 , κ ∈ C , κ ≠ 0 , λ ∈ L 1 ( ∂ D , C ) is a nontrivial complex valued function, and ∂ ν is the outward unit normal vector on ∂ D , relative to D c . Namely, under a certain condition imposed on ( κ , λ ) , we show that if N ≥ 3 and p < p c , where p c = N N − 2 , then the considered problem admits no global weak solutions. However, if N = 2 , then for all p > 1 , the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function.

scholarly journals Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions

2021 ◽  
Vol 6 (11) ◽  
pp. 12182-12224
Author(s):  
Quincy Stévène Nkombo ◽  
◽  
Fengquan Li ◽  
Christian Tathy ◽  
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Radon Measure ◽  
Neumann Boundary ◽  
Large Time Behavior ◽  
Neumann Boundary Conditions ◽  
Degenerate Parabolic Equations ◽  
Measure Data ◽  
Normal Vector ◽  
Time Behavior

<abstract><p>In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data.</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta\psi(u)+h(t)f(x, t) \ \ &amp;\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial\psi(u)}{\partial\eta} = g(u) \ \ &amp;\text{on} \ \ \partial\Omega\times(0, T), \\ u(x, 0) = u_{0}(x) \ \ &amp;\text{in} \ \ \Omega, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ T &gt; 0 $, $ \Omega\subset \mathbb{R}^{N}(N\geq2) $ is an open bounded domain with smooth boundary $ \partial\Omega $, $ \eta $ is an outward normal vector on $ \partial\Omega $. The initial value data $ u_{0} $ is a nonnegative bounded Radon measure on $ \Omega $, the function $ f $ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions $ \psi $, $ g $ and $ h $ satisfy the suitable assumptions.</p></abstract>

scholarly journals Nonexistence of solutions for quasilinear hyperbolic inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Suping Xiao ◽  
Zhong Bo Fang
Keyword(s):  
Weak Solutions ◽  
Function Method ◽  
Source Term ◽  
Cauchy Problems ◽  
Global Weak Solutions ◽  
Test Function ◽  
Singular Source ◽  
Nonexistence Of Solutions ◽  
Test Function Method

AbstractIn this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.

scholarly journals Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Rong Zhang ◽  
Liangchen Wang
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Large Time ◽  
Neumann Boundary ◽  
Rates Of Convergence ◽  
Large Time Behavior ◽  
Global Dynamics ◽  
Time Behavior ◽  
Chemical Signalling ◽  
Chemotaxis System

<p style='text-indent:20px;'>This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t&gt;0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t&gt;0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t&gt;0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ a_1,a_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \chi_1, \chi_2, \chi_3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are positive constants. We first showed some conditions between <inline-formula><tex-math id="M6">\begin{document}$ \frac{\chi_1}{\mu_1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \frac{\chi_2}{\mu_2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\chi_3}{\mu_2} $\end{document}</tex-math></inline-formula> and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.</p>

scholarly journals Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chao Liu ◽  
Bin Liu
Keyword(s):  
Asymptotic Behavior ◽  
Large Time ◽  
Neumann Boundary ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Smooth Bounded Domain ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pursuit Evasion ◽  
Predator Model

<p style='text-indent:20px;'>In this paper, we study the prey-predator model with indirect pursuit-evasion interaction defined on a smooth bounded domain with homogeneous Neumann boundary conditions. We obtain the globa existence and boundedness of the classical solution of the model by estimating <inline-formula><tex-math id="M1">\begin{document}$ L^{p} $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>, and we also show the large time behavior and convergence rate of the solution.</p>

scholarly journals Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2765
Author(s):  
Ravi P. Agarwal ◽  
Soha Mohammad Alhumayan ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Weak Solutions ◽  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Global Weak Solutions ◽  
Test Function ◽  
Test Function Method

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

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