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In this paper, we use a natural iteration technique to prove the existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to unbounded domains, as we show with domains of the form R n − 1 × (− d / 2 , d / 2). [ABSTRACT FROM AUTHOR]

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established. [ABSTRACT FROM AUTHOR]

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for Hartree type equation i u t + u x x = [ K ⁎ | u | 2 ] u , with Cauchy data in modulation spaces M p , q (R). We establish global well-posedness results in M p , p ′ (R) when K (x) = λ | x | γ , (λ ∈ R , 0 < γ < 1) with no smallness condition on initial data, where p ′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature. [ABSTRACT FROM AUTHOR]

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation { − (∇ + i A (x)) 2 u + (λ V (x) + 1) u = (1 | x | μ ⁎ | u | p) | u | p − 2 u x ∈ R N , u ∈ H 1 (R N , C) , where N ≥ 2 , λ > 0 is a real parameter, 0 < μ < 2 , i is the imaginary unit, p ∈ (2 , 2 ⁎ (2 (N − μ) 2 N)) , where 2 ⁎ = 2 N N − 2 if N ≥ 3 , 2 ⁎ = + ∞ , if N = 2. The magnetic potential A ∈ L l o c 2 (R N , R N) and V : R N → R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω 1 , ⋯ , Ω k such that the interior of Ω j is non-empty and ∂ Ω j is smooth, then for λ > 0 large enough, the above equation has at least 2 k − 1 multi-bump solutions. [ABSTRACT FROM AUTHOR]

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 (u 3) x x , u (x , 0) = u 0 (x) ∈ H (R) , where H (R) = W 3 , 1 (R) ∩ H 2 , 2 (R) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale (y , t). In any fixed space-time cone C (y 1 , y 2 , v 1 , v 2) = { (y , t) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , we compute the long time asymptotic expansion of the solution u (x , t) , which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N (I) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second | t | − 1 / 2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O (| t | − 1) from a ∂ ‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X D s − 1 , q (Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation (− Δ) σ u = | x | α u p in B 1 \ { 0 } with an isolated singularity at the origin, where σ ∈ (0 , 1) and the punctured unit ball B 1 \ { 0 } ⊂ R n with n ≥ 2. When − 2 σ < α < 2 σ and n + α n − 2 σ < p < n + 2 σ n − 2 σ , we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (1981) [21] , but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on S + n. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the boundary pointwise C 1 , α and C 2 , α regularity for viscosity solutions of fully nonlinear elliptic equations. That is, if ∂Ω is C 1 , α (or C 2 , α) at x 0 ∈ ∂ Ω , the solution is C 1 , α (or C 2 , α) at x 0. Our results are new even for the Laplace equation. Moreover, our proofs are simple. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term − ∂ x 2 u. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in H s (R 2) for s > 1 / 2 , and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in H s (R 2) for s ≥ 0. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in H s (R 2) for s > − 1 / 2. [ABSTRACT FROM AUTHOR]

Abstract The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic N -function M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large | ξ | and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the periodic discrete nonlinear equation { L u n − ω u n = ± g n ( u n ) , n ∈ Z , lim | n | → ∞ ⁡ u n = 0 , where L is a Jacobi operator, and the nonlinearities g n ( s ) are asymptotically linear as | s | → ∞ . In the two different cases ( ω is a spectral endpoint of L , or it belongs to a finite spectral gap of L ), we obtain the existence of nontrivial solitons of this equation by using variational methods. In particular, a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation. Here, solitons appear when we look for standing waves of some discrete nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]