Semilinear elliptic equations admitting similarity transformations

In this paper we study the equation −Δu+ρ^−(α+2)h(ρ^αu)=0$-\mathrm{\Delta }u+{\rho }^{-(\alpha +2)}h({\rho }^{\alpha }u)=0$ in a smooth bounded domain Ω   where ρ(x)=dist(x,∂Ω)$\rho (x)=\mathrm{dist}(x,\partial \Omega )$, α>0$\alpha >0$ and h is a nondecreasing function which satisfies Keller–Osserman condition. We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on h we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of [13] treating equations of the form −Δu+ρ^βu^q=0$-\mathrm{\Delta }u+{\rho }^{\beta }{u}^q=0$ with q>1$q>1$, β>−2$\beta >-2$.     

Semilinear structural damped waves

We study the Cauchy problem for the semilinear structural damped wave equation with source term with σ ∈ (0,1] in space dimension n ≥ 2 and with a positive constant μ. We are interested in the influence of σ on the critical exponent p_crit in | f(u) | ≈ | u | ^p. This critical exponent is the threshold between global existence in time of small data solutions and blow‐up behavior for some suitable range of p. Our results are optimal for σ = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.     

Semilinear representations of PGL

Let L be the function field of a projective space over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf on is a collection of isomorphisms for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear of degree one is an integral L-tensor power of It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.     

Semilinear Finite Element Method

In the Ritz-Galerkin method the linear subspace of the trial solution is extended to a closed subset. Some results, such as orthogonalization and minimum property of the error function, are obtained. A second order scheme is developed for solving a linear singular perturbation elliptic problem and error estimates are given for a uniform mesh size. Numerical results for linear and semilinear singular perturbation problems are included.     

Singularly Perturbed Semilinear Systems

Solutions of a singularly perturbed vector boundary-value problem are studied under the principal assumption that the trivial solution of the unperturbed equation is stable in certain senses. This is accomplished by constructing special invariant regions in which solutions display the kind of nonuniformity known as boundary-layer behavior.     

Stabilizing Semilinear Parabolic Equations

In this paper we prove the internal feedback stabilization of steady-state solutions of semilinear parabolic equations and introduce a controller synthesis methodology based on finite element approximations of the original PDEs. Numerical tests are given for some one- and two-dimensional nonlinear parabolic equations.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

Semilinear equations on fractal blowups

The paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal, where imbedding of the energy space into Lp is not compact. In absence of invariant transformations involved in conventional concentration-compactness argument, the paper develops convergence reasoning based on the fractal's self-similarity.     

Approximate Controllability for Semilinear Systems

The relations between the (strong) reachable sets of the semilinear evolution equation systems x′(t) + A(t)x(t) = f(t, x(t), u(t)) + Hu(t), x′(t) + A(t)x(t) = f(t, x(t), Hu(t)) + Hu(t) on a Banach space, and their corresponding linear systems are studied. Compared with previous results, the systems considered here are more general (f is not independent of the control u), no compactness assumptions on A or f are imposed in some of our main results, and we suppose f is a set-contraction rather than Lipschitz and have less restriction on the contraction coefficient. Other kinds of conditions are involved to guarantee the approximate controllability.     

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ₁ of (−Δ,H ₀ ¹ (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.     

Singularly Perturbed Semilinear Robin Problems

Several phenomena of current interest in catalytic reaction theory can be phrased in terms of solutions of singularly perturbed, second-order, semilinear differential equations satisfying boundary conditions of Robin type. This paper offers a mathematical theory for such problems in both the scalar and the vector case, and treats some examples involving the pseudo-steady-state hypothesis and a parallel reaction inside of a porous catalyst pellet.     

Semilinear evolution equations in Banach spaces

We study the evolution equation u′(t) = Au(t) + J(u(t)), t ? 0, where e^tA is a C₀ semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K_t: E → E, t > 0, which heuristically are just e^tAJ. Under certain integrability conditions on the K_t we prove existence and uniqueness of local solutions to the integral equation u(t) = e^tAφ + ∝₀^tK_{t ? s}(u(s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K_t, and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in Rⁿ.     

退化的半线性反应扩散方程组解的熄灭

In this paper,the initial boundary value problem of semilinear degenerate reaction-diffusion systems is studied.The regularization method and upper and lower solutions technique are employed to show the existence and continuation of a positive classical solution.The location of quenching points is found.The critical length is estimated by the eigenvalue method.     

Semilinear reaction-diffusion systems of several components

The homogeneous Dirichlet boundary value problem

     

Exact Controllability for Semilinear Wave Equations

In this note, we prove the exact controllability for the semilinear wave equations in any space dimensions under the condition that the nonlinearity behaves like as s → ∞.     

半线性系统的内层现象

本文研究半线性系统
εy″=f(t,y,ε) (ay(a,ε)=A(ε), y(b,ε)=B(ε)
的奇摄动边值问题的内层现象,其中ε>0是一个小参数,y、f、A和B是n-维向量函数。这种向量边值问题似乎还没有研究过,尽管纯量的边值问题已广泛地研究过。在适当地假设下如同纯量问题一样我们得到解的存在、同时使用适当的不等式,这个解的估计也得到。