Superlinear Reaction Research Articles

Superlinear Robin Problems with Indefinite Linear Part

We consider a semilinear Robin problem with an indefinite linear part and a superlinear reaction term, which does not satisfy the usual in such cases AR condition. Using variational methods, together with truncation–perturbation techniques and Morse theory (critical groups), we establish the existence of three nontrivial solutions. Our result extends in different ways the multiplicity theorem of Wang.

Superlinear Robin problems with indefinite linear part

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools we prove two theorems. An existence theorem producing a nontrivial smooth solution and a multiplicity theorem producing a whole unbounded sequence of nontrivial smooth solutions.

Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction

We consider nonlinear Dirichlet and Neumann problems driven by a nonlinear nonhomogeneous differential operator and with a (p−1)-superlinear Carathéodory reaction which does not satisfy the Ambrosetti–Rabinowitz condition. We prove two multiplicity theorems producing two nodal solutions, and answer two open questions posed by Aizicovici et al. (2013) and by Barletta and Papageorgiou (2014). Our approach uses variational methods together with suitable truncation techniques and flow invariance arguments.

Robin problems with a general potential and a superlinear reaction

We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups).

Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter [Formula: see text] varies. We also show the existence of a smallest positive solution [Formula: see text] and investigate the properties of the map [Formula: see text]. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term

Abstract We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is ( p - 1 ) ${{(p-1)}}$ -superlinear near ± ∞ ${\pm\infty}$ without satisfying the Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.

Superlinear Neumann problems with the p-Laplacian plus an indefinite potential

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. First, we prove an existence theorem, and then, under stronger conditions on the reaction, we prove a multiplicity theorem producing three nontrivial solutions. Then, we examine parametric problems with competing nonlinearities (concave and convex terms). We show that for all small values of the parameter \(\lambda >0\), the problem has five nontrivial solutions and if \(p=2\) (semilinear equation), there are six nontrivial solutions. Finally, we prove a bifurcation result describing the set of positive solutions as the parameter \(\lambda >0\) varies.

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.

Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries

We consider a semilinear Dirichlet problem with an unbounded and indefinite potential and a superlinear reaction which need not satisfy the usual, in such cases, Ambrosetti-Rabinowitz condition. Using a combination of variational methods (critical point theory) with truncation and comparison techniques, with Morse theory and with flow invariance arguments, we show that the problem has at least seven nontrivial smooth solutions and provide sign information for all of them.

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

Nonlinear parabolic problems with a very general quadratic gradient term

We study existence and regularity of distributional solutions for a class of nonlinear parabolic problems. The equations we consider have a quasi-linear diffusion operator and a lower-order term, which may grow quadratically in the gradient and may have a very fast growth (for instance, exponential) with respect to the solution. The model problem we refer to is the following \begin{equation} \begin{cases} u_t -\Delta u = \beta(u) |{\nabla} u|^2 +f(x, t), & \ \text{in}\quad \Omega\times (0, T);\\[0.1cm] u(x,t)=0, & \ \text{on}\quad \partial\Omega\times (0, T);\\[0.1cm] u(x, 0)=u_0(x), & \ \text{in}\quad \Omega; \end{cases} \tag*{(0.1)} \label{modello} \end{equation} with $\Omega \subset \mathbb R^N$ a bounded open set, $T>0$, and $\beta(u)\sim e^{|u|}$; as far as the data are concerned, we assume $\exp(\exp(|u_0|))\in L^2({\Omega})$, and $f \in X(0, T; Y({\Omega}))$, where $X, Y$ are Orlicz spaces of logarithmic and exponential type, respectively. We also study a semilinear problem having a superlinear reaction term, a problem that is linked with problem (0.1) by a change of unknown (see (1.3) below). Likewise, we deal with some other related problems, which include a gradient term and a reaction term together.

Single-point blow-up patterns for a nonlinear parabolic equation

We study a nonlinear parabolic equation with a superlinear reaction term. By studying the backward self-similar solutions for this equation, we construct a finite number of self-similar single-point blow-up patterns with different oscillations.

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i). Here we prove that there exist global solutions with slow decay and unbounded free boundary, i.e. of type (ii). Also, we establish uniform a priori estimates for all global solutions. Moreover, we provide a correction to an error in the proof of decay from Ghidouche, Souplet, & Tarzia [5].

An asymptotic and numerical description of self-similar blow-up in quasilinear parabolic equations

We study the blow-up behaviour of two reaction-diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction. We show that in each case the blow-up is self-similar, in contrast to the linear diffusion limit of each in which the diffusion is only approximately self-similar. We then investigate the limit of the self-similar behaviour and describe the transition from a stable manifold blow-up behaviour (for quasilinear diffusion) to a centre manifold one for the linear diffusion.