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The goal of this work is to investigate finite-time blow-up solutions to a class of time-space fractional diffusion equations with nonlinear exponential source terms. In contrast to the small critical data case, which leads to global solutions, we prove in this study that if the initial Schwartz data is large enough, our solutions will blow up in a finite time. The main idea of the analysis is based on the Fourier analytic approach and embeddings between Triebel-Lizorkin spaces and Besov spaces. [ABSTRACT FROM AUTHOR]

For any $ R > 0 $, infinitely many nonradial singular solutions can be constructed for the following equation: $ \begin{equation} -\Delta u = e^u \;\;\; \mbox{in}\; B_R \backslash \{0\} , \;\;\;\;\;\;(0.1)\end{equation} $ where $ B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\} $. To construct nonradial singular solutions, we need to consider asymptotic expansion at the isolated singular point $ x = 0 $ of a prescribed solution of (0.1). Then, nonradial singular solutions of (0.1) can be constructed by using the asymptotic expansion and introducing suitable weighted Hölder spaces.

We consider solutions of some two-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearity of a counterpart of the classical Gauss–Bieberbach–Rademacher second-order equation, which was previously inspected by many authors in connection with problems of the geometry of surfaces with negative Gaussian curvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditions for the absence of a solution in a disk of sufficiently large radius and show that global solutions on the plane can exist only if the coefficient of nonlinearity decays at infinity at the rate at least . Otherwise the mean value of the solution on a circle of radius would tend to with exponential rate as . The Pokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutoff test functions, proves the impossibility of the existence of such global solution. Also, for the solutions in , periodic in all but variables, the absence of global solutions is obtained by similar methods when the nonlinearity coefficient decays at rate slower than . [ABSTRACT FROM AUTHOR]

Given a bounded smooth domain $ \Omega $ in $ \mathbb{R}^2 $, we study the following anisotropic elliptic problem$\begin{cases}-\nabla(a(x) \nabla v)=a(x)\left[e^v-s \phi_1-4 \pi \alpha \delta_q-h(x)\right] & \text { in }\; \Omega, \\ v=0 & \text { on }\; \partial \Omega,\end{cases}$where $ a(x) $ is a positive smooth function, $ s>0 $ is a large parameter, $ h\in C^{0,\gamma}(\overline{\Omega}) $, $ q\in\Omega $, $ \alpha\in(-1,+\infty)\setminus\mathbb{N} $, $ \delta_q $ denotes the Dirac measure with pole at point $ q $ and $ \phi_1 $ is a positive first eigenfunction of the problem $ -\nabla\big(a(x)\nabla \phi\big) = \lambda a(x)\phi $ under Dirichlet boundary condition in $ \Omega $. We show that if $ q $ is both a local maximum point of $ \phi_1 $ and an isolated local maximum point of $ a(x)\phi_1 $, this problem has a family of solutions $ \upsilon_s $ with arbitrary $ m $ bubbles accumulating to $ q $ and the quantity $ \displaystyle{\int}_{\Omega}a(x)e^{\upsilon_s}\rightarrow8\pi(m+1+\alpha)a(q)\phi_1(q) $ as $ s\rightarrow +\infty $, which give a positive answer to the Lazer-McKenna conjecture for this case. [ABSTRACT FROM AUTHOR]

For any , infinitely many nonradial singular solutions can be constructed for the following equation: where . To construct nonradial singular solutions, we need to consider asymptotic expansion at the isolated singular point of a prescribed solution of (0.1). Then, nonradial singular solutions of (0.1) can be constructed by using the asymptotic expansion and introducing suitable weighted Hölder spaces. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R+2 ${\mathbb{R}}_{+}^{2}$ : (−Δ)α2u(x)=|x|aup1(x)eq1v(x),x∈R+2,(−Δ)v(x)=|x|bup2(x)eq2v(x),x∈R+2, $\begin{cases}{\left(-{\Delta}\right)}^{\frac{\alpha }{2}}u\left(x\right)=\vert x{\vert }^{a}{u}^{{p}_{1}}\left(x\right){e}^{{q}_{1}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \\ \left(-{\Delta}\right)v\left(x\right)=\vert x{\vert }^{b}{u}^{{p}_{2}}\left(x\right){e}^{{q}_{2}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \end{cases}$ with Dirichlet boundary conditions, where 0 0. First, we derived the integral representation formula corresponding to the above system under the assumption p1≥−2aα−1 ${p}_{1}\ge -\frac{2a}{\alpha }-1$ . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres.

In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity. [ABSTRACT FROM AUTHOR]

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory. [ABSTRACT FROM AUTHOR]

This article is focused on the existence of a ground state solution to the Kirchhoff problem: - k ∫ Ω | ∇ u | N d x Δ N u = f (x , u) | x | a + λ g (x) in Ω , u = 0 on ∂ Ω , where Ω ⊆ R N (N ≥ 2) is a bounded domain with smooth boundary and a ∈ [ 0 , N) . We assume that f satisfies critical exponential growth at infinity but does not satisfy the well-known Ambrosetti–Rabinowitz condition. We prove the existence of a ground state weak solution via mountain pass theorem and Nehari manifold technique. [ABSTRACT FROM AUTHOR]

In this paper, based on a suitable fractional Trudinger–Moser inequality, we establish sufficient conditions for the existence result of least energy sign-changing solution for a class of one-dimensional nonlocal equations involving logarithmic and exponential nonlinearities. By using a main tool of constrained minimization in Nehari manifold and a quantitative deformation lemma, we consider both subcritical and critical exponential growths. This work can be regarded as the complement for some results of the literature.

In this paper, we are mainly concerned with the following Hartree type equations with exponential nonlinearity (- Δ) u (x) = 1 | x | σ ∗ e pu e p u (x) , in R 2 where p ∈ (0 , + ∞) , u may change sign. We first prove the equivalence between the PDEs and the corresponding integral equations, further to get the exact asymptotic behavior of solutions to the above PDEs equation. Finally, we classify all classical solutions to the integral equations via the method of moving spheres in integral form. Consequently, we obtain the classification results of classical solutions for the PDEs. [ABSTRACT FROM AUTHOR]

In this paper, a new three-dimensional chaotic system with two exponential nonlinearities is presented. The analysis of fixed points of the proposed system suggests existence of one hyperbolic index-2 spiral saddle-type fixed point. The proposed system fulfils Shilnikov criterion and nature of the chaos is found to be dissipative. Bifurcation diagrams, maximum Lyapunov exponents and Kaplan–Yorke dimension are examined through numerical simulations to investigate dynamics of proposed system. The hardware feasibility of the proposed system is illustrated through current feedback operational amplifier (CFOA)-based circuit implementation. The proposed circuit uses six CFOAs, two diodes to establish nonlinearity, eleven resistors and three capacitors. The absence of analog multiplier in the proposed circuit makes it superior to the existing counterpart in the sense that it does not require area and power-consuming active building block. To confirm the chaotic nature of the proposed circuit, LTspice simulations are done to obtain phase portraits which are found to be strange attractors and are topologically different from the shape of the existing attractors. Moreover, we have investigated the synchronization of the proposed chaotic system using adaptive control scheme and proposed CFOA-based complete circuit design of the adaptively synchronized system. Also, the effect of the tolerance of passive components and temperature on the behavior of the proposed chaotic circuit and the complete synchronization circuit has also been studied. It is found that the circuit is sensitive to the value of resistors and temperature to an extent and can work properly within their limits. [ABSTRACT FROM AUTHOR]

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: { − a (∫ Ω | ∇ u | N d x) Δ N u = f (x , u) | x | b + λ h (x) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N (N ≥ 2) and 0 ≤ b < N. Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: − a (∫ R N | ∇ u | N + V (x) | u | N d x) (Δ N u + V (x) | u | N − 2 u) = g (x , u) | x | b + λ h (x) in R N , where N ≥ 2 and 0 ≤ b < N. We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively.

This paper aims to establish the existence of a weak solution for the nonlocal problem: −a∫ΩH(x,|∇u|)dxΔHu=f(x,u)inΩ,u=0on∂Ω,$$ \left\{\begin{array}{ll}-a\left(\underset{\Omega}{\int}\mathcal{H}\left(x,|\nabla u|\right) dx\right){\Delta}_{\mathcal{H}}u& =f\left(x,u\right)\kern0.30em \mathrm{in}\kern0.30em \Omega, \\ {}u& =0\kern0.30em \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\end{array}\right. $$where Ω⊆ℝN,N≥2$$ \Omega \subseteq {\mathbb{R}}^N,N\ge 2 $$ is a bounded and smooth domain containing two open and connected subsets Ωp$$ {\Omega}_p $$ and ΩN$$ {\Omega}_N $$ such that Ω¯p∩Ω¯N=∅$$ {\overline{\Omega}}_p\cap {\overline{\Omega}}_N=\varnothing $$ and ΔHu=div(h(x,|∇u|)∇u)$$ {\Delta}_{\mathcal{H}}u=\mathit{\operatorname{div}}\left(h\left(x,|\nabla u|\right)\nabla u\right) $$ is the H$$ \mathcal{H} $$‐Laplace operator. We assume that ΔH$$ {\Delta}_{\mathcal{H}} $$ reduces to Δp(x)$$ {\Delta}_{p(x)} $$ in Ωp$$ {\Omega}_p $$ and to ΔN$$ {\Delta}_N $$ in ΩN$$ {\Omega}_N $$, the nonlinear function f:Ω×ℝ→ℝ$$ f:\Omega \times \mathbb{R}\to \mathbb{R} $$ acts as |t|p∗(x)−2t$$ {\left|t\right|}^{p^{\ast }(x)-2}t $$ on Ωp$$ {\Omega}_p $$ and as eα|t|N/(N−1)$$ {e}^{\alpha {\left|t\right|}^{N/\left(N-1\right)}} $$ on ΩN$$ {\Omega}_N $$ for sufficiently large |t|$$ \mid t\mid $$. To establish the existence results in a Musielak–Sobolev space, we use a variational technique based on the mountain pass theorem. [ABSTRACT FROM AUTHOR]

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some $ L^p-L^q $-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in $ L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N) $, the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space.

In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P)−Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞,- \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0}\mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively. [ABSTRACT FROM AUTHOR]

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate on the radial solutions in an annulus. First, we construct a time-local solution with an abstract theory of differential equations. Next, we show that decreasing energy exists in this problem. Finally, we establish a global solution for the sufficiently small initial value and parameter by Sobolev embedding and Poincaré inequalities together with some technical estimates. Moreover, when we take the smaller parameter, we prove that the global solution tends to zero as time goes to infinity. [ABSTRACT FROM AUTHOR]

This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed the solution's lower and upper bound of the blow-up time for the equation. Our results complement the existing results. [ABSTRACT FROM AUTHOR]

We study regularity properties of weak solutions to the boundary value problem for the equation âˆ'Î" ρ + au = f in a bounded domain Ω ⊂ R N , where ρ = e âˆ' div | ∇ u | p âˆ' 2 ∇ u + β 0 | ∇ u | âˆ' 1 ∇ u . This problem is derived from the mathematical modelling of crystal surfaces. It is known that the exponential term can be a measure-valued function. In this paper we obtain a partial regularity result, which asserts that there exists an open subset Ω0 ⊂ Ω such that |Ω\Ω0| = 0 and the exponential term is locally bounded in Ω0. Furthermore, if x 0 ∈ Ω\Ω0, then ρ vanishes of N + 2 âˆ' É› order at x 0 for each É› ∈ (0, 2). [ABSTRACT FROM AUTHOR]

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution. [ABSTRACT FROM AUTHOR]

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation -\Delta u + {\rm e}^u({\rm e}^u-1) =\mu\quadin \Omega with the Dirichlet boundary condition. Approximating $\mu$ by a sequence $(\mu_n)_{n \in\mathbb N}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in\mathbb{N}$, we are interested in establishing the convergence of the sequence $(u_n)_{n\in\mathbb{N}}$ to a function $u^#$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^#$.

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity. [ABSTRACT FROM AUTHOR]

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some L p − L q -type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in L p ( R N) ∩ L ∞ ( R N) , the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space. [ABSTRACT FROM AUTHOR]

This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with stable solutions to the fractional elliptic equation (- Δ) s u = e u in R N , where (- Δ) s is the fractional Laplacian with 0 < s < 1 . By developing a new technique with non-compactly supported test functions, we establish the nonexistence of stable solutions provided that N < 10 s . This result is optimal when s ↑ 1 . On the other hand, we believe that our technique can be used to study stable solutions of elliptic equations/systems involving the fractional Laplacian. [ABSTRACT FROM AUTHOR]

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂ t u − Δ e − Δ u + e − Δ u − 1 = 0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6]. Our investigations reveal that we must control the size of both ‖ e − Δ u (x , 0) ‖ W 2 , 2 (Ω) and ‖ e Δ u (x , 0) ‖ ∞ , Ω suitably to achieve our results. Related results in [8,10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonclassical diffusion equations on ℝN with hereditary memory, in presence of singularly oscillating external forces depending on a positive parameter ε and a new class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. Under a general assumption on the memory kernel κ and for a very large class of nonlinearities, we prove the existence of uniform attractors in H1(ℝN)×Lμ2(ℝ+,H1(ℝN)). The uniform boundedness (w.r.t. ε) and the convergence of uniform attractors Aε as ε tends to 0 are also studied. Our results extend and improve some results in Anh and Toan. [ABSTRACT FROM AUTHOR]

In this article, we study the existence, multiplicity, regularity and asymptotic behavior of the positive solutions to the problem of half-Laplacian with singular and exponential growth nonlinearity in one dimension (see below (P λ)). We prove two results regarding the existence and multiplicity of solutions to the problem (P λ). In the first result, existence and multiplicity have been proved for classical solutions via bifurcation theory while in the latter result multiplicity has been proved for critical exponential nonlinearity by variational methods. An independent question of symmetry and monotonicity properties of classical solution has been answered in the paper. To characterize the behavior of large solutions, we further study isolated singularities for the singular semi linear elliptic equation in Ω ⊂ R n , n ⩾ 2 s involving exponential growth nonlinearities in the more general framework of (− Δ) s operator and for all 0 < s < 1 (see below (P s)). [ABSTRACT FROM AUTHOR]

In this paper, the global existence and nonexistence of solutions for a Klein-Gordon equation, appearing in a variety of physical situations, with exponential type source term and supercritical initial energy (E(0) > d) are investigated in a bounded domain . In the framework of potential well, a functional including both of initial data is defined and by sign invariance of this functional the global existence of weak solutions in the case of high initial energy is proved. Moreover, under some conditions imposed on initial displacement and initial velocity a finite time blow up result is provided which extends a result given in the literature. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein–Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space H 1 (ℝ) × L 2 (ℝ). We prove that any energy solution has a global bound of the L t , x 6 space-time norm, and hence, scatters in H 1 (ℝ) × L 2 (ℝ) as t → ± ∞. The proof is based on the argument by Killip–Stovall–Visan (Trans. Amer. Math. Soc.364(3) (2012) 1571–1631). However, since well-posedness in H 1 / 2 (ℝ) × H − 1 / 2 (ℝ) for NLKG with the exponential nonlinearity holds only for small initial data, we use the L t 6 W x s − 1 / 2 , 6 -norm for some s > 1 2 instead of the L t , x 6 -norm, where W x s , p denotes the s th order L p -based Sobolev space. [ABSTRACT FROM AUTHOR]

An initial-boundary value problem for the multidimensional equation of ion-sound waves in a plasma is considered. Its time-local solvability in the classical sense in Hölder spaces is proved. This is a development of results in our previous papers, where the local solvability of one-dimensional analogs of the equation under consideration was established and, in the general case (regardless of the dimension of the space), sufficient conditions for the blow-up of the solution were obtained. [ABSTRACT FROM AUTHOR]

We study radial symmetry of entire solutions of the equation 0.1 $$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$ It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called 'maximal radial entire solution', and infinitely many radial entire solutions with the latter behaviour, which are called 'non-maximal radial entire solutions'. The necessary and sufficient conditions for an entire solution u of (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solution u of (0.1) to be a non-maximal radial entire solution. [ABSTRACT FROM AUTHOR]

The focus of the present work is a one-dimensional system of weakly coupled degenerate semi-linear parabolic equations of optimal portfolio in a regime-switching with power utility function, suggested in Valdez and Vargiolu [Optimal portfolio in a regime-switching model, in Proceedings of the Ascona '11 Seminar on Stochastic Analysis, Random Fields and Applications, R.C. Dalang, M. Dozzi, and F. Russo, eds., 2013, pp. 435–449]. We extend the model, deriving a new linear parabolic system for indifference option valuation. Then, we investigate the properties of the solution of the systems. Further, we develop and analyse an adequate, flux limited finite difference discretization, which preserves the typical features of the differential problem solution. Numerical experiments are presented and discussed. [ABSTRACT FROM AUTHOR]

We consider the infinite multiplicity of entire solutions for the elliptic equation Δ u + K (x) e u + μ f (x) = 0 in ℝ n , n ⩾ 3. Under suitable conditions on K and f , the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like | x |ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log | x | near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following boundary value problem {−Δu + u = λ|x − q1|R2α1⋯|x−qn|2αnup−1eup, u > 0, in Ω; ∂u/∂ν = 0 on ∂Ω, where Ω is a bounded domain in R2 with smooth boundary, points q1,...,qn ∈ Ω a1,...,an ∈ (0,∞)∖ N, λ > 0 is a small parameter, 0

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = Δu + eu in ℝN. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9, and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen (2008) in a finite ball to the whole space. [ABSTRACT FROM AUTHOR]

In this paper, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension 1 ≤ d ≤ 3. Our proofs base on the analyticity of Mittag-Leffler functions, the framework of prior estimates and the type of exponential nonlinearity. [ABSTRACT FROM AUTHOR]

In this note we prove the nonexistence of stable solutions to the p-Laplace equation $-\Delta_p u = e^u$ on the entire Euclidean space $\mathbb{R}^N$, where $p>2$ and $N < \frac{p(p+3)}{p-1}$.

Let Ω be a bounded domain in with smooth boundary, and let 𝓧1; 𝓧2; · · ·, 𝓧m be points in Ω. We are concerned with the singular stationary non-homogenous q-Kuramoto-Sivashinsky eaquation (q-KSE:

This paper proposes a numerical method to deal with the two-dimensional hyperbolic equations with nonlocal integral conditions. The nonlocal integral equation usually is of major challenge in the frame work of the numerical solutions of partial differential equations. The method benefits from collocation radial basis function method, the generalized thin plate splines (GTPS) radial basis functions are used. Therefore, it does not require any struggle to determine shape parameter (In other RBFs, it is time- consuming step). The present technique is one of the truly meshless methods in where it does not require any background integration cells over local or global domains and it is in contrast to weak form methods in where all integrations are carried out locally or globally over quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. The obtained results for some numerical examples reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems. [ABSTRACT FROM AUTHOR]

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity i ∂ t u + Δ u = u | x | b (e α | u | 2 − 1) ; (t , x) ∈ ℝ × ℝ 2 , where 0 < b < 1 and α = 2 π (2 − b). We establish local and global well-posedness in the subcritical and critical regimes. [ABSTRACT FROM AUTHOR]

Abstract: We prove a sharp Liouville type theorem for stable W l o c 1 , p solutions of the equation − Δ p u = f ( x ) e uon the entire Euclidean space R N, where p > 2 and f is a continuous and nonnegative function in R N ∖ { 0 } such that f ( x ) ≥ a | x | q as | x | ≥ R 0 > 0, where q > − p and a > 0. Our theorem holds true for 2 ≤ N < p 2 + 3 p + 4 q p − 1 and is sharp in the case f ( x ) = a | x | q. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in R 2 with smooth boundary, we study the following anisotropic elliptic problem − ∇ ( a ( x ) ∇ υ ) = a ( x ) [ e υ − s ϕ 1 − h ( x ) ] in Ω , υ = 0 on ∂ Ω , where a ( x ) is a positive smooth function, h ∈ C 0 , α ( Ω ¯ ) , s > 0 is a large parameter and ϕ 1 is a positive first eigenfunction of the problem − ∇ ( a ( x ) ∇ ϕ ) = λ a ( x ) ϕ under Dirichlet boundary condition in Ω . We construct solutions of this problem which exhibit multiple concentration behavior around maximum points of a ( x ) ϕ 1 in the domain as s → + ∞ . [ABSTRACT FROM AUTHOR]

We consider the semilinear elliptic equation Δu+K(|x|)eu=0 in RN for N>2, and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For N=10+4ℓ with ℓ>-2, the equation changes its nature drastically according to the sign of the derivative of r-ℓK(r) when r-ℓK(r) is monotonic in r and r-ℓK(r)→1 as r→∞. [ABSTRACT FROM AUTHOR]

Bifurcation for prescribed mean curvature equations in one dimension has been intensively investigated in recent years, and a striking phenomenon discovered is that the length of the interval may affect the shapes of bifurcation curves. However, to our best knowledge, no such results are known in higher dimensions. In this paper, we study a class of prescribed mean curvature equations in bounded domains of R N with general nonlinearities f satisfying f ( 0 ) = 0 and f ′ ( 0 ) > 0 . We establish some formulas for directions of bifurcation at simple eigenvalues, which lead to a sufficient and necessary condition to ensure that the directions depend on the size of the domain. In contrast, this phenomenon does not occur for the semilinear case. Some interesting examples, including logistic and perturbed exponential nonlinearities, are also investigated. [ABSTRACT FROM AUTHOR]

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger{Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem in-volving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the N-Laplacian operator. [ABSTRACT FROM AUTHOR]

We obtain global well-posedness, scattering, and global Lt4Hx1,4 spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation ... in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao-Visan-Zhang from [25]. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson [12] and the work of [15] and [14] to prove corresponding spacetime estimates which imply scattering. [ABSTRACT FROM AUTHOR]