Electronic Journal of Differential Equations
https://digital.library.txstate.edu/handle/10877/86
2019-05-21T01:32:37ZQuasi-Geostrophic Type Equations with Weak Initial Data
https://digital.library.txstate.edu/handle/10877/7949
Quasi-Geostrophic Type Equations with Weak Initial Data
Wu, Jiahong
We study the initial value problem for the quasi-geostrophic type equations ∂θ / ∂t + u • ∇θ + (-Δ)(λ)θ = 0, on ℝ(n) x (0, ∞), θ(x,0) = θ(0)(x), x ∈ ℝ(n), where λ(0 ≤ λ ≤ 1) is a fixed parameter and u = (u(j)) is divergence free and determined from θ through the Riesz transform u(j) = ±R(π)(j)θ, with π(j) a permutation of 1,2, •••, n. The initial data θ(0) is taken in the Sobolev space Ĺ(r),(p) with negative indices. We prove local well-posedness when 1/2 < λ ≤ 1, 1 < p < ∞, n/p ≤ 2λ - 1, r = n/p - (2λ - 1) ≤ 0. We also prove that the solution is global if θ(0) is sufficiently small.
1998-06-12T00:00:00ZBranching of Periodic Orbits from Kukles Isochrones
https://digital.library.txstate.edu/handle/10877/7948
Branching of Periodic Orbits from Kukles Isochrones
Toni, Bourama
We study local bifurcations of limit cycles from isochronous (or linearizable) centers. The isochronicity has been determined using the method of Darboux linearization, which provides a birational linearization for the examples that we analyze. This transformation simplifies the analysis by avoiding the complexity of the Abelian integrals appearing in other approaches. As an application of this approach, we show that the Kukles isochrone (linear and nonlinear) has at most one branch point of limit cycles. Moreover, for each isochrone, there are small perturbations with exactly one continuous family of limit cycles.
1998-05-13T00:00:00ZSymmetry and Convexity of Level Sets of Solutions to the Infinity Laplace's Equation
https://digital.library.txstate.edu/handle/10877/7947
Symmetry and Convexity of Level Sets of Solutions to the Infinity Laplace's Equation
Rosset, Edi
We consider the Dirichlet problem -Δ∞u = f(u) in Ω, u = 0 on ∂Ω, where Δ∞u = u(x)(i), u(x)(j), u(x)(i)(x)j) and f is a nonnegative continuous function. We investigate whether the solutions to this equation inherit geometrical properties from the domain Ω. We obtain results concerning convexity of level sets and symmetry of solutions.
1998-12-09T00:00:00ZExistence and Multiplicity of Solutions to a p-Laplacian Equation with Nonlinear Boundary Condition
https://digital.library.txstate.edu/handle/10877/7946
Existence and Multiplicity of Solutions to a p-Laplacian Equation with Nonlinear Boundary Condition
Pfluger, Klaus
We study the nonlinear elliptic boundary value problem
Au = f(x,u) in Ω, Bu = g(x,u) on ∂Ω, where A is an operator of p-Laplacian type, Ω is an unbounded domain in ℝ(N) with non-compact boundary, and f and g are subcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both f and g are superlinear. Also we show existence of at least two nonnegative solutions when one of the two functions f, g is sublinear and the other one is superlinear. The proofs are based on variational methods applied to weighted function spaces.
1998-04-10T00:00:00Z