http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
SOLVABILITY OF NONLINEAR ELLIPTIC TYPE EQUATION WITH TWO UNRELATED NON STANDARD GROWTHS
Sert, Ugur,Soltanov, Kamal Korean Mathematical Society 2018 대한수학회지 Vol.55 No.6
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths $$-div\({\mid}{\nabla}u{\mid}^{p_1(x)-2}{\nabla}u\)-\sum\limits^n_{i=1}D_i\({\mid}u{\mid}^{p_0(x)-2}D_iu\)+c(x,u)=h(x),\;{\in}{\Omega}$$ in a bounded domain ${\Omega}{\subset}{\mathbb{R}}^n$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
Solvability of nonlinear elliptic type equation with two unrelated non standard growths
Ugur Sert,Kamal Soltanov 대한수학회 2018 대한수학회지 Vol.55 No.6
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths \begin{equation*} -div\left(\left\vert\nabla u\right\vert^{p_{1}\left( x\right)-2}\nabla u\right)-\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p_{0}\left( x\right)-2}D_{i}u\right)+c\left( x,u\right) \!=\!h\left( x\right) ,~x\!\in\! \Omega \end{equation*} in a bounded domain $\Omega \subset \mathbb{R}^{n}$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
ON SOLVABILITY OF A CLASS OF DEGENERATE KIRCHHOFF EQUATIONS WITH LOGARITHMIC NONLINEARITY
Ugur Sert Korean Mathematical Society 2023 대한수학회지 Vol.60 No.3
We study the Dirichlet problem for the degenerate nonlocal parabolic equation u<sub>t</sub> - a(||∇u||<sup>2</sup><sub>L<sup>2</sup>(Ω)</sub>)∆u = C<sub>b</sub> ||u||<sup>β</sup><sub>L<sup>2</sup>(Ω)</sub> |u|<sup>q(x,t)-2</sup> u log |u| + f in Q<sub>T</sub>, where Q<sub>T</sub> := Ω × (0, T), T > 0, Ω ⊂ ℝ<sup>N</sup>, N ≥ 2, is a bounded domain with a sufficiently smooth boundary, q(x, t) is a measurable function in Q<sub>T</sub> with values in an interval [q<sup>-</sup>, q<sup>+</sup>] ⊂ (1, ∞) and the diffusion coefficient a(·) is a continuous function defined on ℝ<sub>+</sub>. It is assumed that a(s) → 0 or a(s) → ∞ as s → 0<sup>+</sup>, therefore the equation degenerates or becomes singular as ||∇u(t)||2 → 0. For both cases, we show that under appropriate conditions on a, β, q, f the problem has a global in time strong solution which possesses the following global regularity property: ∆u ∈ L<sup>2</sup>(Q<sub>T</sub>) and a(||∇u||<sup>2</sup><sub>L<sup>2</sup>(Ω)</sub>)∆u ∈ L<sup>2</sup>(Q<sub>T </sub>).
Existence and Behavior Results for a Nonlocal Nonlinear Parabolic Equation with Variable Exponent
Sert, Ugur,Ozturk, Eylem Department of Mathematics 2020 Kyungpook mathematical journal Vol.60 No.1
In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.