We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in (Ann Inst H Poincaré Anal Non Linéaire, 4(1):1–47 1987). We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

We study the long-time dynamics of the Navier–Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey–Sobolev spaces in such a general system, then any Leray–Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.

This paper establishes Calderon-Zygmund type regularity estimates for solutions of the conormal derivative problem for a class of linear elliptic systems in divergence-form with singular, degenerate coefficients in bounded domains. In our class of equations, the principal terms are fourth order tensors of measurable functions that behave as some weight function in the Muckenhoupt class of A2-weights. Regularity estimates for gradient of weak solutions in weighted Lebesgue spaces are established under some natural smallness conditions on the mean oscillation of coefficients. The results obtained recover known results when the coefficients are uniformly elliptic. These results can be considered as the Sobolev counterparts of the classical Holder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class. Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation. A class of degenerate coefficients satisfying the smallness condition is characterized. A counter example to demonstrate the necessity of the smallness condition on the coefficients is given. Our $W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of the Hölder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982.

We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers converging to infinity.

We give necessary and sufficient conditions for the existence of weak solutions to the model equation-Delta(p)u = sigma u(q) on R-nin the case 0 < q < p - 1, where sigma >= 0 is an arbitrary locally integrable function, or measure, and Delta(p)u = div(del u vertical bar del u vertical bar(p-2)) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation-Delta(p)v = b vertical bar del v vertical bar(p)/v + sigma on R-n,where b > 0. These results are new even in the classical case p = 2.Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, del u), fractional Lapincians (-Delta)(alpha), or fully nonlinear k-Hessian operators. (C) 2016 Elsevier Inc. All rights reserved.

We study quasilinear elliptic equations of the type-Delta(p)u = sigma u(q) in R-n,where Delta(p)u = del center dot (del u vertical bar del u vertical bar(p-2)) is the p-Laplacian (or a more general A-Laplace operator div A(x, del u)), 0 < q < p - 1, and sigma >= 0 is an arbitrary locally integrable function or measure on R-n. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian (-Delta)(alpha) for 0 < alpha < n/2 are treated as well, including explicit estimates for radially symmetric sigma. Our results are new even in the classical case p = 2 and alpha = 1. (C) 2016 Elsevier Ltd. All rights reserved.

We give an explicit necessary and sufficient condition for the existence of a finite energy solution to the quasilinear elliptic equation -Δ _pu=σ u^q, u ≥ 0 in ℝ^n, where Δ _p is the p -Laplacian, p>1 , and σ≥ 0 is an arbitrary locally integrable function (or measure) on ℝ^n , in the case 0<q < p-1 (below the “natural growth” rate q=p-1 ). We also prove that such a solution is unique. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the A -Laplacian divA(x,∇· ) in place of Δ _p are considered as well.