On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

Existence of solutions for a nonlinear PDE with an inverse square potential

Via Linking theorem and delicate energy estimates, the existence of nontrivial solutions for a nonlinear PDE with an inverse square potential and critical sobolev exponent is proved. This result gives a partial (positive) answer to an open problem proposed in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494).     

Gevrey class regularity for quasi-linear sub-elliptic equations of second order

In this work we study the Gevrey regularity of solutions to a general class of second order quasi-linear equations. Under some kind of sub-ellipticity conditions, we obtain the Gevrey regularity of weak solutions to these equations.     

Extremal solutions for nonlinear parabolic problems with discontinuities

This paper examines nonlinear parabolic initial-boundary value problems with a discontinuous forcing term, which is locally of bounded variation. Assuming that there exist an upper solution and a lower solution , we prove the existence of a maximal and of a minimal solution within the order interval [,] L ^P (P xZ). Our approach is based on a Jordan-type decomposition for the discontinuous forcing term and on a fixed point theorem for nondecreasing maps in ordered Banach spaces.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

Weighted estimates of solutions to second order parabolic equations

We evaluate the rate of decay for solutions to second order parabolic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. Our approach is based on a special growth lemma, and it works for both divergence and non-divergence equations, in domains satisfying a general “exterior measure condition” (A). The result for elliptic case is published in Cho and Safonov (2007) [2].     

Homogenization and penalization of functional first-order PDE

In this paper we study the asymptotic behavior of viscosity solutions for a functional partial differential equation with a small parameter as the parameter tends to zero. We study simultaneous effects of homogenization and penalization in functional first-order PDE. We establish a convergence theorem in which the limit equation is identified with some first order PDE.     

Persistence of solutions to higher order nonlinear Schrödinger equation

Applying an Abstract Interpolation Lemma, we showed persistence of solutions of the initial value problem to higher order nonlinear Schrödinger equation, also called Airy-Schrödinger equation, in weighted Sobolev spaces X2,θ, for θ∈[0,1].     

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p$p$-Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ$\lambda$ is greater than the second eigenvalue _λ2${\lambda }_2$ of the negative p$p$-Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p$p$-Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.     

Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups

This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems in divergence form in Carnot groups. The technique of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context. We establish Caccioppoli type inequalities and partial regularity with optimal local Hölder exponents for horizontal gradients of weak solutions to systems under super-quadratic natural structure conditions and super-quadratic controllable structure conditions, respectively.     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

Local existence of solutions to dissipative nonlinear evolution equations with mixed types

In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types

     

Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order

In this paper, the asymptotic behavior of the solution to the initial–boundary value problem for a nonlinear evolution equation of fourth order

equation(1)

_utt−_a1uxx−_a2uxxt−_a3uxxtt=φ_(ux)x$u_{tt}-a_1{u}_{xx}-a_2{u}_{xxt}-a_3{u}_{xxtt}=\phi {\left(u_x\right)}_x$

is studied. The sufficient conditions for blow-up of the solutions to the initial–boundary value problems for Eq. (1) are given.     

Large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions

We study the dissipation of solutions of the Cauchy problem for the nonlinear dissipative wave equation in odd multi-spatial dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygens principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675