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Communications on Pure & Applied Analysis

2005 , Volume 4 , Issue 3

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Existence of periodic solution for a class of systems involving nonlinear wave equations
Claudianor O. Alves
2005, 4(3): 487-498 doi: 10.3934/cpaa.2005.4.487 +[Abstract](131) +[PDF](203.2KB)
In this paper, we show the existence of periodic solution for a class of systems involving nonlinear wave equations. The main tool used is the dual method.
Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates
D. Bartolucci and L. Orsina
2005, 4(3): 499-522 doi: 10.3934/cpaa.2005.4.499 +[Abstract](171) +[PDF](381.5KB)
We analyze the singular behavior of the Green's function for uniformly elliptic equations on smooth and bounded two dimensional domains. Then, we are able to generalize to the uniformly elliptic case some sharp estimates for Liouville type equations due to Brezis-Merle [7] and, in the same spirit of [3], a "mass" quantization result due to Y.Y. Li [21]. As a consequence, we obtain uniform a priori estimates for solutions of the corresponding Dirichlet problem. Then, we improve the standard existence theorem derived by direct minimization and, in the same spirit of [17] and [37], obtain the existence of Mountain Pass type solutions.
Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary
C. Brändle, F. Quirós and Julio D. Rossi
2005, 4(3): 523-536 doi: 10.3934/cpaa.2005.4.523 +[Abstract](128) +[PDF](245.0KB)
We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.
Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation
Qiang Du, Chun Liu, R. Ryham and X. Wang
2005, 4(3): 537-548 doi: 10.3934/cpaa.2005.4.537 +[Abstract](197) +[PDF](1560.4KB)
In this paper, we study the effects of the spontaneous curvature on the static deformation of a vesicle membrane under the elastic bending energy, with prescribed bulk volume and surface area. Generalizing the phase field models developed in our previous works, we deduce a new energy formula involving the spontaneous curvature effects. Several axis-symmetric configurations are obtained through numerical simulations. Some analysis on the effects of the spontaneous curvature on the vesicle membrane shapes are also provided.
Existence and non-existence for a mean curvature equation in hyperbolic space
Elias M. Guio and Ricardo Sa Earp
2005, 4(3): 549-568 doi: 10.3934/cpaa.2005.4.549 +[Abstract](111) +[PDF](281.5KB)
There exists a well-known criterion for the solvability of the Dirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classical result was established by Serrin in 1969. Focusing the Dirichlet Problem for radial vertical graphs P.-A. Nitsche has established an existence and non-existence results on account of a criterion based on the notion of a hyperbolic cylinder. In this work we carry out a similar but distinct result in hyperbolic space considering a different Dirichlet Problem based on another system of coordinates. We consider a non standard cylinder generated by horocycles cutting orthogonally a geodesic plane $\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We prove that a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$ of this cylinder along $\partial \Omega$ and the prescribed mean curvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forall y\in\partial\Omega$ yields existence of our Dirichlet Problem. Thus we obtain existence of surfaces whose graphs have prescribed mean curvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribed boundary data $\varphi.$ This result is sharp because if our condition fails at a point $y$ a non-existence result can be inferred.
Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals
Damiano Foschi
2005, 4(3): 569-588 doi: 10.3934/cpaa.2005.4.569 +[Abstract](93) +[PDF](296.6KB)
We discuss the asymptotic behaviour for the best constant in $L^p$-$L^q$ estimates for trigonometric polynomials and for an integral operator which is related to the solution of inhomogeneous Schrödinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.
On certain nonlinear parabolic equations with singular diffusion and data in $L^1$
C. García Vázquez and Francisco Ortegón Gallego
2005, 4(3): 589-612 doi: 10.3934/cpaa.2005.4.589 +[Abstract](65) +[PDF](346.9KB)
An existence result is established for a class of nonlinear parabolic equations having a coercive diffusion matrix blowing-up for a finite value of the unknown, a second hand $f\in L^1(Q)$, and an initial data $u_0\in L^1(\Omega)$. We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some uniqueness results are also shown under additional hypotheses.
New dissipated energies for the thin fluid film equation
Richard S. Laugesen
2005, 4(3): 613-634 doi: 10.3934/cpaa.2005.4.613 +[Abstract](88) +[PDF](338.5KB)
The thin fluid film evolution $h_t = -(h^n h_{x x x})_x$ is known to conserve the fluid volume $\int h dx$ and to dissipate the "energies" $\int h^{1.5-n} dx$ and $\int h_x^2 dx$. We extend this last result by showing the energy $\int h^p h_x^2 dx$ is dissipated for some values of $p < 0$, when $\frac{1}{2} < n < 3$. For example when $n=1$, the Hele-Shaw equation $h_t = -(h h_{x x x})_x$ dissipates $\int h^{-1/2} h_x^2 dx$.
Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid
Yuming Qin, T. F. Ma, M. M. Cavalcanti and D. Andrade
2005, 4(3): 635-664 doi: 10.3934/cpaa.2005.4.635 +[Abstract](100) +[PDF](322.7KB)
This paper is concerned with the exponential stability of solutions in $H^4$ for the Navier--Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains $\Omega_n$ in $\mathbb{R}^n (n=2,3)$.
A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions
Satoshi Kosugi, Yoshihisa Morita and Shoji Yotsutani
2005, 4(3): 665-682 doi: 10.3934/cpaa.2005.4.665 +[Abstract](215) +[PDF](282.4KB)
We consider the Ginzburg-Landau equation with a positive parameter, say lambda, and solve all equilibrium solutions with periodic boundary conditions. In particular we reveal a complete bifurcation diagram of the equilibrium solutions as lambda increases. Although it is known that this equation allows bifurcations from not only a trivial solution but also secondary bifurcations as lambda varies, the global structure of the secondary branches was open. We first classify all the equilibrium solutions by considering some configuration of the solutions. Then we formulate the problem to find a solution which bifurcates from a nontrivial solution and drive a reduced equation for the solution in terms of complete elliptic integrals involving useful parametrizations. Using some relations between the integrals, we investigate the reduced equation. In the sequel we obtain a global branch of the bifurcating solution.
Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions
Zhenhua Zhang
2005, 4(3): 683-693 doi: 10.3934/cpaa.2005.4.683 +[Abstract](119) +[PDF](214.8KB)
This paper is concerned with the asymptotic behavior of solutions to the phase-field equations subject to the Neumann boundary conditions where a Lojasiewicz-Simon type inequality plays an important role. In this paper, convergence of the solution of this problem to an equilibrium, as time goes to infinity, is proved.

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