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1534-0392
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Communications on Pure & Applied Analysis
July 2013 , Volume 12 , Issue 4
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2013, 12(4): i-v
doi: 10.3934/cpaa.2013.12.4i
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Igor V. Skrypnik, a prominent Ukrainian mathematician, was born on November 13, 1940, in Zhmerinka, Ukraine. He graduated from the Lviv State University in 1962 and defended his Candidate-Degree thesis entitled $A$-Harmonic Forms on Riemannian Spaces in 1965. His teacher and supervisor was Yaroslav B. Lopatinskii who played an important role in the formation of mathematical interests of young Skrypnik. In 1965--1967, Skrypnik worked as an assistant, senior lecturer and assistant professor at the Lviv University.
Igor V. Skrypnik, a prominent Ukrainian mathematician, was born on November 13, 1940, in Zhmerinka, Ukraine. He graduated from the Lviv State University in 1962 and defended his Candidate-Degree thesis entitled $A$-Harmonic Forms on Riemannian Spaces in 1965. His teacher and supervisor was Yaroslav B. Lopatinskii who played an important role in the formation of mathematical interests of young Skrypnik. In 1965--1967, Skrypnik worked as an assistant, senior lecturer and assistant professor at the Lviv University.
2013, 12(4): 1527-1546
doi: 10.3934/cpaa.2013.12.1527
+[Abstract](3093)
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The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: \begin{eqnarray} u_{t}=div(a(x,t,u)|u|^{\alpha(x,t)}|\nabla u|^{p(x,t)-2} \nabla u) +f(x,t) \end{eqnarray} with given variable exponents $\alpha(x,t)$ and $p(x,t)$. We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.
The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: \begin{eqnarray} u_{t}=div(a(x,t,u)|u|^{\alpha(x,t)}|\nabla u|^{p(x,t)-2} \nabla u) +f(x,t) \end{eqnarray} with given variable exponents $\alpha(x,t)$ and $p(x,t)$. We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.
2013, 12(4): 1547-1568
doi: 10.3934/cpaa.2013.12.1547
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In this article we study quasilinear systems of two types, in a domain $\Omega$ of $R^N$ : with absorption terms, or mixed terms: \begin{eqnarray} (A): \mathcal{A}_{p} u=v^{\delta}, \mathcal{A}_{q}v=u^{\mu},\\ (M): \mathcal{A}_{p} u=v^{\delta}, -\mathcal{A}_{q}v=u^{\mu}, \end{eqnarray} where $\delta$, $\mu>0$ and $1 < p$, $ q < N$, and $D = \delta \mu- (p-1) (q-1) > 0$; the model case is $\mathcal{A}_p = \Delta_p$, $\mathcal{A}_q = \Delta_q.$ Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type: \begin{eqnarray} u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leq Cd(x,\partial\Omega)^{-\frac{q(p-1)+p\mu}{D}}. \end{eqnarray} Concerning system $(M)$, we show that $v$ always satisfies Harnack inequality. In the case $\Omega=B(0,1)\backslash \{0\}$, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.
In this article we study quasilinear systems of two types, in a domain $\Omega$ of $R^N$ : with absorption terms, or mixed terms: \begin{eqnarray} (A): \mathcal{A}_{p} u=v^{\delta}, \mathcal{A}_{q}v=u^{\mu},\\ (M): \mathcal{A}_{p} u=v^{\delta}, -\mathcal{A}_{q}v=u^{\mu}, \end{eqnarray} where $\delta$, $\mu>0$ and $1 < p$, $ q < N$, and $D = \delta \mu- (p-1) (q-1) > 0$; the model case is $\mathcal{A}_p = \Delta_p$, $\mathcal{A}_q = \Delta_q.$ Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type: \begin{eqnarray} u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leq Cd(x,\partial\Omega)^{-\frac{q(p-1)+p\mu}{D}}. \end{eqnarray} Concerning system $(M)$, we show that $v$ always satisfies Harnack inequality. In the case $\Omega=B(0,1)\backslash \{0\}$, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.
2013, 12(4): 1569-1585
doi: 10.3934/cpaa.2013.12.1569
+[Abstract](2372)
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We prove uniqueness of weak (or entropy) solutions for nonmonotone elliptic equations of the type \begin{eqnarray} -div (a(x,u)\nabla u)=f \end{eqnarray} in a bounded set $\Omega\subset R^N$ with Dirichlet boundary conditions. The novelty of our results consists in the possibility to deal with cases when $a(x,u)$ is only Hölder continuous with respect to $u$.
We prove uniqueness of weak (or entropy) solutions for nonmonotone elliptic equations of the type \begin{eqnarray} -div (a(x,u)\nabla u)=f \end{eqnarray} in a bounded set $\Omega\subset R^N$ with Dirichlet boundary conditions. The novelty of our results consists in the possibility to deal with cases when $a(x,u)$ is only Hölder continuous with respect to $u$.
2013, 12(4): 1587-1633
doi: 10.3934/cpaa.2013.12.1587
+[Abstract](2192)
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We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.
We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.
2013, 12(4): 1635-1656
doi: 10.3934/cpaa.2013.12.1635
+[Abstract](2687)
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We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.
We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.
2013, 12(4): 1657-1686
doi: 10.3934/cpaa.2013.12.1657
+[Abstract](3047)
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We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals $\Gamma$-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.
We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals $\Gamma$-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.
2013, 12(4): 1687-1703
doi: 10.3934/cpaa.2013.12.1687
+[Abstract](2768)
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The paper presents some pieces from algebra, theory of function and differential geometry, which have emerged in frames of the modern theory of fully nonlinear second order partial differential equations and revealed their interdependence. It also contains a survey of recent results on solvability of the Dirichlet problem for m-Hessian equations, which actually brought out this development.
The paper presents some pieces from algebra, theory of function and differential geometry, which have emerged in frames of the modern theory of fully nonlinear second order partial differential equations and revealed their interdependence. It also contains a survey of recent results on solvability of the Dirichlet problem for m-Hessian equations, which actually brought out this development.
2013, 12(4): 1705-1729
doi: 10.3934/cpaa.2013.12.1705
+[Abstract](2415)
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We investigate the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given in the literature.
We investigate the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given in the literature.
2013, 12(4): 1731-1744
doi: 10.3934/cpaa.2013.12.1731
+[Abstract](2784)
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For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
2013, 12(4): 1745-1753
doi: 10.3934/cpaa.2013.12.1745
+[Abstract](1675)
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We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
2013, 12(4): 1755-1768
doi: 10.3934/cpaa.2013.12.1755
+[Abstract](2421)
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We study long-time behavior for the Cauchy problem of degenerate parabolic system which in the scalar case coincides with classical porous media equation. Sharp bounds of the decay in time estimates of a solution and its size of support were established. Moreover, local space-time estimates under the optimal assumption on initial data were proven.
We study long-time behavior for the Cauchy problem of degenerate parabolic system which in the scalar case coincides with classical porous media equation. Sharp bounds of the decay in time estimates of a solution and its size of support were established. Moreover, local space-time estimates under the optimal assumption on initial data were proven.
2013, 12(4): 1769-1782
doi: 10.3934/cpaa.2013.12.1769
+[Abstract](2056)
+[PDF](413.8KB)
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Let $\Gamma=\{\gamma(\tau)\in R^N\times [0,T], \gamma(0)=(0,0)\}$ be $C^{0,1}$ -- space-time curve and continuos function $h(x,t)>0$ in $ R^N\times [0,T]\setminus \Gamma (h(x,t)=0$ on $\Gamma$). We investigate the behaviour as $k\to \infty$ of the fundamental solutions $u_k$ of equation $u_t-\Delta u+h(x,t)u^p=0$, $p>1$, satisfying singular initial condition $u_k(x,0)=k\delta_0$. The main problem is whether the limit $u_\infty$ is still a solution of the above equation with isolated point singularity at $(0,0)$, or singularity set of $u_\infty$ contains some part or all $\Gamma$.
Let $\Gamma=\{\gamma(\tau)\in R^N\times [0,T], \gamma(0)=(0,0)\}$ be $C^{0,1}$ -- space-time curve and continuos function $h(x,t)>0$ in $ R^N\times [0,T]\setminus \Gamma (h(x,t)=0$ on $\Gamma$). We investigate the behaviour as $k\to \infty$ of the fundamental solutions $u_k$ of equation $u_t-\Delta u+h(x,t)u^p=0$, $p>1$, satisfying singular initial condition $u_k(x,0)=k\delta_0$. The main problem is whether the limit $u_\infty$ is still a solution of the above equation with isolated point singularity at $(0,0)$, or singularity set of $u_\infty$ contains some part or all $\Gamma$.
2013, 12(4): 1783-1812
doi: 10.3934/cpaa.2013.12.1783
+[Abstract](2225)
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We prove the existence and uniqueness of global solutions to the Cauchy problem for a class of parabolic equations of the p-Laplace type. In the singular case $p<2$ there are no restrictions on the behaviour of solutions and initial data at infinity. In the degenerate case $p>2$ we impose a restriction on growth of solutions at infinity to obtain global existence and uniqueness. This restriction is given in terms of weighted energy classes with power-like weights.
We prove the existence and uniqueness of global solutions to the Cauchy problem for a class of parabolic equations of the p-Laplace type. In the singular case $p<2$ there are no restrictions on the behaviour of solutions and initial data at infinity. In the degenerate case $p>2$ we impose a restriction on growth of solutions at infinity to obtain global existence and uniqueness. This restriction is given in terms of weighted energy classes with power-like weights.
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