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Start Over Subject existence and uniqueness

1. Existence of positive solution of a singular partial differential equation

Creator:
Zhang, Shuqin
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
mixed Riemann-Liouville fractional derivative, function space concerning fractional derivative, existence and uniqueness, positive solution, and fixed point theorem
Language:
English
Description:
Motivated by Vityuk and Golushkov (2004), using the Schauder Fixed Point Theorem and the Contraction Principle, we consider existence and uniqueness of positive solution of a singular partial fractional differential equation in a Banach space concerning with fractional derivative.
Rights:
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2. On solutions of quasilinear wave equations with nonlinear damping terms

Creator:
Park, Jong Yeoul and Bae, Jeong Ja
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
quasilinear wave equation, existence and uniqueness, asymptotic behavior, and Galerkin method
Language:
English
Description:
In this paper we consider the existence and asymptotic behavior of solutions of the following problem: \[ u_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2 +\beta \Vert \nabla v(t,x)\Vert _2^2)\Delta u(t,x) +\delta |u_t(t,x)|^{p-1}u_t(t,x) \quad =\mu |u(t,x)|^{q-1}u(t,x), \quad x \in \Omega ,\quad t \ge 0, v_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2+ \beta \Vert \nabla v(t,x)\Vert _2^2) \Delta v(t,x) +\delta |v_t(t,x)|^{p-1}v_t(t,x) \quad =\mu |v(t,x)|^{q-1}v(t,x), \quad x \in \Omega ,\quad t \ge 0, u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega , v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega , u|_{_{\partial \Omega }}=v|_{_{\partial \Omega }}=0 \] where $q > 1$, $ p \ge 1$, $ \delta >0$, $ \alpha > 0$, $ \beta \ge 0 $, $\mu \in \mathbb R $ and $\Delta $ is the Laplacian in $\mathbb R^N$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type

Creator:
Jong Yeoul and Bae, Jeong Ja
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
existence and uniqueness, Galerkin method, and nondegenerate wave equation
Language:
English
Description:
Let $\Omega $ be a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac{\partial ^2 y}{\partial t^2}-M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \Delta y -\frac{\partial }{\partial t}\Delta y=f(y) \quad \text{in} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text{in} \Sigma _1=\Gamma _{\!1} \times (0,\infty ), M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \frac{\partial y}{\partial \nu } +\frac{\partial }{\partial t}\Bigl (\frac{\partial y}{\partial \nu }\Bigr )=g \quad \text{in} \Sigma _0=\Gamma _{\!0} \times (0,\infty ), y(0)=y_0,\quad \frac{\partial y}{\partial t}\,(0)=y_1 \quad \text{in} \quad \Omega , \qquad \mathrm{(1)}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations

Creator:
Bravyi, E,, Hakl, Robert, and Lomtatidze, Alexander
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
linear functional differential equations, Cauchy problem, existence and uniqueness, and differential inequalities
Language:
English
Description:
Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem \[ u^{\prime }(t)=\ell (u)(t)+q(t), \qquad u(a)=c, \] where $\ell \:C(I,\mathbb R)\rightarrow L(I,\mathbb R)$ is a linear bounded operator, $q\in L(I,\mathbb R)$, and $c\in \mathbb R$, are established.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public