اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExistence and non-existence results for a class of semilinear nonlocal operators with exterior condition
69
0
0.0
(
0
)
تأليف
Anup Biswas
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
قيم البحث
اقرأ أيضاً
We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Delta u = (1-u) u^m - lambda u^n$ in a bounded domain $Omega subset mathbb{R}^N$ endowed with
the zero Dirichlet boundary data, where $0<m leq 1$ and $n>0$. When $lambda > 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.
In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 le p<1+frac{2}{N}$, begin{equation*} begin{cases} u_t+v cdot abla u-Delta u=|u|^p-int_{mathbb T^N} |u|^p quad & textrm{on} quad mathbb T^N, u
textrm{periodic} quad & textrm{on} quad partial mathbb T^N end{cases} end{equation*} with initial data $u_0$ defined on $mathbb T^N$. Here $v$ is an incompressible flow, and $mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.
In this paper, we study a one dimensional nonlinear equation with diffusion $- u(-partial_{xx})^{frac{alpha}{2}}$ for $0leq alphaleq 2$ and $ u>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(mathbb{
R})cap H^{1/2}(mathbb{R})$ when $0leqalphaleq 2$. For subcritical $1<alphaleq 2$ and critical case $alpha=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<alphaleq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $alpha=1$, if the initial data $rho_0$ satisfies $- u<infrho_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If $rho_0geq0$, the solution exists globally and converges to steady state.
In this paper we address the analytical investigation of a model for adhesive contact, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the un
ilateral contact conditions, the physical constraints on the internal variables, as well as the integral contributions related to the nonlocal forces. For the associated initial-boundary value problem we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.
A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functio
ns, assuming the collision kernel $K$ to be given by $K(x,y)= x^{alpha} y^{beta} + x^{beta} y^{alpha}$ with $alpha le beta le 1$. When $alpha + beta in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $alpha + beta in [0,1)$ and $alpha ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $alpha <0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
