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Search: All articles in the CJM digital archive with keyword existence

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1. CJM Online first

Du, Zhuoran; Fang, Yanqin; Gui, Changfeng
A class of degenerate elliptic equations with nonlinear boundary conditions
We consider positive solutions of the problem \begin{equation} (*)\qquad \left\{ \begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n}, \\ \frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n}, \\ \end{array} \right. \end{equation} where $a\in (-1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and $\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case $b=0$, it is associated to fractional Laplacian equation $(-\Delta)^{s}u=u^{q} $ in entire space $\mathbb{R}^{n-1}$. We obtain the existence of positive axially symmetric solutions to ($*$) for the case $a\in (-1,0)$ in $n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a-2}, \;\;q\geq\frac{n-a}{n+a-2}$. The nonexistence is obtained for the case $a\in (-1,0)$, $b\geq 0$ and any $p,~q\gt 1$ in $n=2$ as well.

Keywords:existence, non-existence, positive solutions, degenerate elliptic equation, nonlinear boundary conditions, symmetry, monotonicity
Categories:35D30, 35J70, 35J25

2. CJM Online first

Saanouni, Tarek
Global and non global solutions for some fractional heat equations with pure power nonlinearity
The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

Keywords:nonlinear fractional heat equation, global Existence, decay, blow-up
Category:35Q55

3. CJM 2016 (vol 68 pp. 521)

Emamizadeh, Behrouz; Farjudian, Amin; Zivari-Rezapour, Mohsen
Optimization Related to Some Nonlocal Problems of Kirchhoff Type
In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton we are able to show that both problems are solvable, and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions. The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type, which is stable. Some numerical results are included to confirm the theory.

Keywords:Kirchhoff equation, rearrangements of functions, maximization, existence, optimality condition
Categories:35J20, 35J25

4. CJM 2012 (vol 64 pp. 1415)

Selmi, Ridha
Global Well-Posedness and Convergence Results for 3D-Regularized Boussinesq System
Analytical study to the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solution with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter $\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, Arselà-Ascoli compactness theorem and a Friedrichs like approximation scheme.

Keywords:regularizing Boussinesq system, existence and uniqueness of weak solution, convergence results, compactness method in frequency space
Categories:35A05, 76D03, 35B40, 35B10, 86A05, 86A10

5. CJM 2002 (vol 54 pp. 1121)

Bao, Jiguang
Fully Nonlinear Elliptic Equations on General Domains
By means of the Pucci operator, we construct a function $u_0$, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma
Categories:35D05, 35D10, 35J60, 35J67

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