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1937-1632
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1937-1179
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Discrete & Continuous Dynamical Systems - S
August 2014 , Volume 7 , Issue 4
Issue on nonlinear elliptic and parabolic partial differential equations
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2014, 7(4): i-i
doi: 10.3934/dcdss.2014.7.4i
+[Abstract](1596)
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Abstract:
The study of elliptic and parabolic nonlinear partial differential equation has manifold aspects and can be seen from a number of different perspectives and points of view. The mathematical tools involved can, similarly, be quite different since they may include, for example, functional analysis, calculus of variations, topological techniques, geometric analysis, semigroup theory, numerical methods.
It would therefore be hopeless to give a complete picture of the research going on nowadays on these topics. Nevertheless, we tried to collect in the present volume several contributions of some leading scholars in these fields, hoping that they could give some hints of some of the main research lines in the field, identifying in particular some setting in which the elliptic theory is an important clue to analyze the parabolic situation, and viceversa.
In the papers collected here, all of which have been anonymously refereed as requested by the high standards of this Journal, several rapidly developing and important topics are discussed and developed.
Among them, we mention the following ones: two-phases free boundary problems; qualitative properties of solutions of Lane-Emden-Fowler equations; Hopf fibration and singularly perturbed elliptic equations; biharmonic elliptic boundary value problems; positivity preserving issues in models for clamped plates; Liouville theorems for Hardy-Littlewood-Sobolev systems; regularity of solutions of degenerate elliptic equations; decay properties for degenerate parabolic problems; porous media and fast diffusion equations driven by fractional Laplacians; fine asymptotics of solutions to the fast diffusion equation; well-posedness of nonlinear integral equations with general kernels and asymptotics of the corresponding solutions; local lower and upper bounds for solutions to doubly nonlinear singular parabolic equations; nonlinear initial value problems that model evolution and selection in living systems in connection with kinetic theory; relationships between optimal inequalities and nonlinear flows; generalized solutions, via non-archimedean fields, to equations which may not have solutions in distributional sense; models for oscillations in suspension bridges.
The success of this collection depends on the quality of the papers and of the high reputation of the Authors: we are grateful to all the contributors of the present volume.
The study of elliptic and parabolic nonlinear partial differential equation has manifold aspects and can be seen from a number of different perspectives and points of view. The mathematical tools involved can, similarly, be quite different since they may include, for example, functional analysis, calculus of variations, topological techniques, geometric analysis, semigroup theory, numerical methods.
It would therefore be hopeless to give a complete picture of the research going on nowadays on these topics. Nevertheless, we tried to collect in the present volume several contributions of some leading scholars in these fields, hoping that they could give some hints of some of the main research lines in the field, identifying in particular some setting in which the elliptic theory is an important clue to analyze the parabolic situation, and viceversa.
In the papers collected here, all of which have been anonymously refereed as requested by the high standards of this Journal, several rapidly developing and important topics are discussed and developed.
Among them, we mention the following ones: two-phases free boundary problems; qualitative properties of solutions of Lane-Emden-Fowler equations; Hopf fibration and singularly perturbed elliptic equations; biharmonic elliptic boundary value problems; positivity preserving issues in models for clamped plates; Liouville theorems for Hardy-Littlewood-Sobolev systems; regularity of solutions of degenerate elliptic equations; decay properties for degenerate parabolic problems; porous media and fast diffusion equations driven by fractional Laplacians; fine asymptotics of solutions to the fast diffusion equation; well-posedness of nonlinear integral equations with general kernels and asymptotics of the corresponding solutions; local lower and upper bounds for solutions to doubly nonlinear singular parabolic equations; nonlinear initial value problems that model evolution and selection in living systems in connection with kinetic theory; relationships between optimal inequalities and nonlinear flows; generalized solutions, via non-archimedean fields, to equations which may not have solutions in distributional sense; models for oscillations in suspension bridges.
The success of this collection depends on the quality of the papers and of the high reputation of the Authors: we are grateful to all the contributors of the present volume.
2014, 7(4): 593-616
doi: 10.3934/dcdss.2014.7.593
+[Abstract](2037)
+[PDF](501.2KB)
Abstract:
This paper deals with a new kind of generalized functions, called ``ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper.
This paper deals with a new kind of generalized functions, called ``ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper.
2014, 7(4): 617-629
doi: 10.3934/dcdss.2014.7.617
+[Abstract](2114)
+[PDF](207.9KB)
Abstract:
We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following \begin{equation} \label{prob1} \left\{ \begin{array}{lll} \displaystyle u_t - {\rm div} \left( \frac{\nabla u}{(1+|u|)^2} \right) = 0, & \hbox{in} & \Omega_T; \\ u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\ u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega. \end{array} \right. \end{equation} We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.
We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following \begin{equation} \label{prob1} \left\{ \begin{array}{lll} \displaystyle u_t - {\rm div} \left( \frac{\nabla u}{(1+|u|)^2} \right) = 0, & \hbox{in} & \Omega_T; \\ u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\ u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega. \end{array} \right. \end{equation} We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.
2014, 7(4): 631-652
doi: 10.3934/dcdss.2014.7.631
+[Abstract](1880)
+[PDF](567.5KB)
Abstract:
We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.
We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.
2014, 7(4): 653-671
doi: 10.3934/dcdss.2014.7.653
+[Abstract](2412)
+[PDF](420.7KB)
Abstract:
We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on $R^N$ . Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Hénon-Lane-Emden system.
We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on $R^N$ . Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Hénon-Lane-Emden system.
2014, 7(4): 673-693
doi: 10.3934/dcdss.2014.7.673
+[Abstract](2510)
+[PDF](2518.0KB)
Abstract:
We present some recent progress on the analysis of two-phase free boundary problems governed by elliptic operators, with non-zero right hand side. We also discuss on several open questions, object of future investigations.
We present some recent progress on the analysis of two-phase free boundary problems governed by elliptic operators, with non-zero right hand side. We also discuss on several open questions, object of future investigations.
2014, 7(4): 695-724
doi: 10.3934/dcdss.2014.7.695
+[Abstract](2535)
+[PDF](953.3KB)
Abstract:
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
2014, 7(4): 725-735
doi: 10.3934/dcdss.2014.7.725
+[Abstract](2241)
+[PDF](387.8KB)
Abstract:
We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation $u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.
We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation $u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.
2014, 7(4): 737-760
doi: 10.3934/dcdss.2014.7.737
+[Abstract](1950)
+[PDF](269.0KB)
Abstract:
In this paper we show some properties regarding the local behaviour of local weak solutions to a class of doubly nonlinear singular parabolic equations.
In this paper we show some properties regarding the local behaviour of local weak solutions to a class of doubly nonlinear singular parabolic equations.
2014, 7(4): 761-766
doi: 10.3934/dcdss.2014.7.761
+[Abstract](2153)
+[PDF](471.6KB)
Abstract:
It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a clamped plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a clamped plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
2014, 7(4): 767-783
doi: 10.3934/dcdss.2014.7.767
+[Abstract](2460)
+[PDF](264.7KB)
Abstract:
We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
2014, 7(4): 785-791
doi: 10.3934/dcdss.2014.7.785
+[Abstract](2621)
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Abstract:
We first review some history prior to the failure of the Tacoma Narrows suspension bridge. Then we consider some popular accounts of this in the popular physics literature, and the scientific and scholarly basis for these accounts and point out some failings. Later, we give a quick introduction to three different models, one single particle, one a continuum model, and two systems with two degrees of freedom.
We first review some history prior to the failure of the Tacoma Narrows suspension bridge. Then we consider some popular accounts of this in the popular physics literature, and the scientific and scholarly basis for these accounts and point out some failings. Later, we give a quick introduction to three different models, one single particle, one a continuum model, and two systems with two degrees of freedom.
2014, 7(4): 793-805
doi: 10.3934/dcdss.2014.7.793
+[Abstract](2158)
+[PDF](443.7KB)
Abstract:
We study the asymptotic behaviour as $p\rightarrow \infty$ of the nodal radial solutions $u_p$ of the problem \begin{equation*} \left\{ \begin{array}{rlll} -\Delta u&=&|u|^{p-1}u& \text{in }\Omega \\ u&=&0& \text{on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is an annulus in $\mathbb{R}^N$, $N\geq 2$. We also analyze the spectrum of the linearized operator associated to $u_p$ in the case when $u_p$ has only two nodal regions. In particular, we prove that the Morse index of $u_p$ tends to $\infty$ as $p$ goes to $\infty$.
We study the asymptotic behaviour as $p\rightarrow \infty$ of the nodal radial solutions $u_p$ of the problem \begin{equation*} \left\{ \begin{array}{rlll} -\Delta u&=&|u|^{p-1}u& \text{in }\Omega \\ u&=&0& \text{on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is an annulus in $\mathbb{R}^N$, $N\geq 2$. We also analyze the spectrum of the linearized operator associated to $u_p$ in the case when $u_p$ has only two nodal regions. In particular, we prove that the Morse index of $u_p$ tends to $\infty$ as $p$ goes to $\infty$.
2014, 7(4): 807-821
doi: 10.3934/dcdss.2014.7.807
+[Abstract](2175)
+[PDF](407.0KB)
Abstract:
This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.
This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.
2014, 7(4): 823-838
doi: 10.3934/dcdss.2014.7.823
+[Abstract](2351)
+[PDF](513.4KB)
Abstract:
In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.
In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.
2014, 7(4): 839-855
doi: 10.3934/dcdss.2014.7.839
+[Abstract](2053)
+[PDF](4362.5KB)
Abstract:
We address the question, for which $\lambda \in \mathbb{R}$ is the boundary value problem \begin{equation*} \left\{ \begin{array}{cc} \Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\ u=\Delta u=0 & \text{on }\partial \Omega , \end{array} \right. \end{equation*} positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what happens, when $\lambda $ passes the maximal value for which positivity is preserved.
We address the question, for which $\lambda \in \mathbb{R}$ is the boundary value problem \begin{equation*} \left\{ \begin{array}{cc} \Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\ u=\Delta u=0 & \text{on }\partial \Omega , \end{array} \right. \end{equation*} positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what happens, when $\lambda $ passes the maximal value for which positivity is preserved.
2014, 7(4): 857-885
doi: 10.3934/dcdss.2014.7.857
+[Abstract](4586)
+[PDF](592.6KB)
Abstract:
We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
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