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In this paper, the following Kirchhoff type elliptic equation \[ -\left(a+b\int_{\Omega}|\nabla u|^2\,\mathrm{d}x\right)\Delta u= \lambda|u|^{q-2}u\ln |u|^2+|u|^4u, \] − (a + b ∫ Ω | ∇u | 2 d x) Δu = λ | u | q − 2 uln ⁡ | u | 2 + | u | 4 u , which involves a power type nonlinearity with critical Sobolev exponent and a logarithmic type perturbation, is investigated. The fact that the logarithmic term satisfies neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition brings some additional difficulties when one is looking for weak solutions to this problem. By the use of the Mountain Pass Lemma and the concentration compactness principle, it is proved that the problem admits a ground state solution for $ q\in (4,6) $ q ∈ (4 , 6) and for any $ \lambda >0 $ λ > 0. It is worth pointing out that the analysis includes both the nondegenerate case (a>0) and the degenerate case (a = 0). [ABSTRACT FROM AUTHOR]

In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall's inequalities and Log-Sobolev inequalities play an important role. [ABSTRACT FROM AUTHOR]

In this paper, two results regarding the existence and multiplicity of ground state solutions for a fractional Kirchhoff-type system involving logarithmic nonlinearities are obtained via variational methods. The proposed problem is motivated by several mathematical models that arise, for example, in quantum mechanics, nuclear physics, quantum optics, transport and diffusion phenomena, effective quantum gravity, open quantum systems, the theory of superfluidity, and Bose–Einstein condensation. The first result provides the existence of a ground state solution for the proposed problem. Under a different set of hypotheses with respect to the first result, a second one is obtained, which provides the existence of at least two non-trivial ground state solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Kirchhoff‐type pseudo parabolic equation driven by a nonlocal integro‐differential operator ℒK$$ {\mathcal{L}}_K $$: ut|x|2s+M([u]s2)ℒKu+ℒKut=|u|p−2ulog|u|,$$ \frac{u_t}{{\left|x\right|}^{2s}}+M\left({\left[u\right]}_s^2\right){\mathcal{L}}_Ku+{\mathcal{L}}_K{u}_t={\left|u\right|}^{p-2}u\log \mid u\mid, $$where [u]s$$ {\left[u\right]}_s $$ represents the Gagliardo seminorm of u$$ u $$. Instead of imposing specific assumptions on the Kirchhoff function, we introduce a more general sense to establish the local existence of weak solutions. Moreover, via the sharp fractional Hardy inequality, the decay estimates for global weak solutions, the blow‐up criterion, blow‐up rate, and the upper and lower bounds of the blow‐up time are derived. Lastly, we discuss the global existence and finite time blow‐up results with high initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we study the following logarithmic fractional Kirchhoff equation: $$\begin{align*} &\left(a+b \int_{\mathbb{R}^{3}}|(-\Delta)^{s/2} u|^{2} \,{\rm d} x\right)(-\Delta)^{s} u+V(x) u\notag \\ &\quad =|u|^{p-2} u \log u^{2}+\lambda|u|^{q-2} u, \quad{\rm in} \ \mathbb{R}^{3}, \end{align*}$$ (a + b ∫ R 3 | (− Δ) s / 2 u | 2 d x) (− Δ) s u + V (x) u = | u | p − 2 ulog ⁡ u 2 + λ | u | q − 2 u , in R 3 , where a, b>0, $ (-\Delta)^{s} $ (− Δ) s is the fractional Laplace operator with $ q \geq ~2_{s}^{*}=\frac {6}{3-2 s} $ q ≥ 2 s ∗ = 6 3 − 2 s and V satisfies suitable conditions. Since the presence of supercritical exponents leads to a lack of compactness, we introduce a truncation function to reduce the exponents, allowing compactness to be recovered. We show that the above equation has a ground state solution and a sign-changing solution via the constraint variation method, the quantitative deformation lemma and Moser's iterative method. [ABSTRACT FROM AUTHOR]

This article concerns a connection between the fractional Schrödinger equation and the logarithmic fractional Schrödinger equation. By rescaling and the constrained minimization method, we prove the asymptotic behaviors of normalized ground states for two equations. [ABSTRACT FROM AUTHOR]

begin{equation*} \begin{cases} ( - \Delta )_{p}^{s} u +|u|^{ p-2} u= \lambda h_1(x) |u|^{ \theta-2}u\ln |u| + \dfrac{q}{q+r}b_1(x)|v|^{r}|u|^{q-2}u & \mbox{in} \ \Omega,\\ ( - \Delta )_{p}^{t} v + |v|^{ p-2} v= \mu h_2(x) |v|^{\theta-2}v\ln |v| + \dfrac{r }{q+r}b_2(x)|u|^{r}|v|^{q-2}v & \mbox{in} \ \Omega,\\ u=v=0, & \mbox{in} \ \mathbb{R}^N \setminus \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s, t \in (0,1)$, $ N > \max\{ps, pt\} $, $\lambda, \mu > 0$, $p \leq \theta$, $2 < q + r < \min\big\{\frac{pN}{N-ps}, \frac{pN}{N-pt}\big\}$, and the additional weights $h_1, h_2, b_1, b_2 \in C(\overline{\Omega})$ are such that: $b_1(x), b_2(x)$ are positive functions and $h_1(x), h_2(x)$ are sign-changing functions. The operators $(-\Delta)_p^s$ and $(-\Delta)_p^t$ represents, both, fractional $p$-Laplacian operator, a generalization for the fractional Laplacian $(-\Delta)^s, 0< s < 1$ $(p = 2)$, defined in a integral way as \begin{equation*} (-\Delta)^s u(x) := \dfrac{c(n,s)}{2} \int_{\mathbb{R}^N}\dfrac{2u(x) - (x+y)-u(x-y)}{|y|^{n+2s}}dy,\qquad x \in \mathbb{R}^N, \end{equation*} where $c(n,s)$ is a positive normalizing constant, and another fractional operator. Specifically, the operators $(-\Delta)_p^s$ and $(-\Delta)_p^t$ are defined, up to a normalization constant, by the formula \begin{equation*} (-\Delta)^{\ell}_{p} u(x) := \lim_{\varepsilon \rightarrow 0^{+}} 2 \int_{\mathbb{R}^N \setminus B_{\varepsilon}(x)} \frac{|u(x) - u(y)|^{p-2} (u(x) - u(y))}{|x-y|^{N+sp}} dy, \end{equation*} for all $u \in C^{\infty}_{0}(\mathbb{R}^N), x \in \mathbb{R}^N,$ and $ \ell \in \{ s,t\}.$

In this paper, we investigate the existence of ground state solution to a class of Schrödinger equation involving logarithmic nonlinearity. To overcome the lack of smoothness, the corresponding functional $J$ is first decomposed into the sum of a $\mathcal{C}^{1}$ functional and a convex lower semicontinuous functional by adapting to the approach of Squassina–Szulkin in [Calc. Var. Partial Differential Equations 54(2015), 585–597]. Secondly, the existence of a ground state solution to the studied equation is proved by using the Mountain Pass Theorem under the weakened Ambrosetti–Rabinowitz conditions.

ABSTRACT In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. and Han. We firstly estimate the upper bound function with doubly exponential property for the weak solutions and establish the doubly exponential decay and at most doubly exponential growth criteria for the weak solutions by the logarithmic Sobolev inequality, the derivative formula for the product, the Newton–Leibniz formula, and the logarithmic properties. We secondly establish a newly infinity blow‐up criterion such that the lower bound function of weak solutions is at least doubly exponential growth by the nonconcavity method and the nonincreasing property for the energy functional. We finally establish a sufficient condition such that the upper bound function with doubly exponential growth is greater than or equal to the lower bound function with doubly exponential growth by the quotient comparison principle. In addition, we also find the threshold of the doubly exponential growth and the doubly exponential decay and obtain ‖u‖22→0+$$ {\left\Vert u\right\Vert}_2^2\to {0}^{+} $$ for any t≥0$$ t\ge 0 $$ as ‖u0‖22→0+.$$ {\left\Vert {u}_0\right\Vert}_2^2\to {0}^{+}. $$ These research results improve previous results from both decay and growth. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for a singular parabolic $ p $-biharmonic equation with gradient-type logarithmic nonlinearity. By the cut-off technique, combined with the methods of Faedo-Galerkin approximation and multiplier, we establish the local solvability. Further, by virtue of the family of potential wells and the modified differential inequality technique, we derive the threshold results of global existence and nonexistence with low initial energy conditions ($ J(u_{0})\leq d $), the decay property, blow-up with negative initial energy, the estimates of blow-up rate, and the bounds of life span of blow-up solutions with subcritical initial energy. Furthermore, we adopt the Hardy-Sobolev inequality to show the finite time blow-up result with arbitrary initial energy. [ABSTRACT FROM AUTHOR]

We consider the following logarithmic Schrödinger–Poisson system: - Δ u + ϕ u = u p - 2 u ln u 2 + λ f (x , u) , in Ω , - Δ ϕ = u 2 , in Ω , ϕ , u = 0 , on ∂ Ω , where Ω is a bounded domain in R 3 with smooth boundary ∂ Ω , p ∈ (4 , 6) , f(x, u) is continuous without any other condition. Using constrained variational method, Mountain Pass Theorem and iterative technique, we prove the existence of mountain pass solutions when λ > 0 small enough. Moreover, with f (x , 0) ≠ 0 in Ω , the above system possesses the another local minimum nontrivial solution. Finally, we prove that for any j ∈ N , there exists λ j > 0 , such that if 0 < λ < λ j , the above system possesses at least j distinct high energy solutions. [ABSTRACT FROM AUTHOR]

We construct the global existence for a wave equation involving the fractional Laplacian with a logarithmic nonlinear source by using the Galerkin approximations. The corresponding results for equations with classical Laplacian are considered as particular cases of our assertions.

Abstract This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations.

This article is devoted to the global existence and extinction behavior of the weak solution to a fast diffusion pp-Laplace equation with logarithmic nonlinearity and special medium void. By applying energy estimates approach, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, the global existence result is proved and the sufficient conditions on the occurrence of the extinction and nonextinction phenomena are given.

We investigate the following logarithmic Kirchhoff-type equation: \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}+V(x)u^{2}dx\right)[-\Delta u+V(x)u]=|u|^{p-2}u\ln |u|,\qquad x\in\mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $4

In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: \begin{equation*} \begin{cases} \epsilon^{2s}\left(-\Delta\right)^{s} u+V(x)u-\phi(x)u= u \log {u^{2}}&\quad\text{ in }\mathbb{R}^{3}, \\ \epsilon^{2t}\left(-\Delta\right)^{t}\phi=|u|^{2}&\quad\text{ in }\mathbb{R}^{3}, \end{cases} \end{equation*} where $\epsilon>0$, $s,t\in(0,1)$, $\left(-\Delta\right)^{\alpha}$ is the fractional Laplacian and $V$ is a saddle-like potential.

In this paper, we considered a singular parabolic -Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations. [ABSTRACT FROM AUTHOR]

We construct the global existence for a wave equation involving the fractional Laplacian with a logarithmic nonlinear source by using the Galerkin approximations. The corresponding results for equations with classical Laplacian are considered as particular cases of our assertions. [ABSTRACT FROM AUTHOR]

This research delves into a comprehensive investigation of a class of ℑ -Hilfer generalized fractional nonlinear equation originated from a capillarity phenomenon involving a logarithmic nonlinearity and Dirichlet boundary conditions. The nonlinearity of the problem, in general, do not satisfies the Ambrosetti-Rabinowitz type condition. Using critical point theorem with variational approach and the (S +) property of the operator, we establish the existence of positive solutions of our problem with respect to every positive parameter ξ in appropriate ℑ -fractional spaces. Our main results is novel and its investigation will enhance the scope of the literature on differential equation of ℑ -Hilfer fractional generalized capillary phenomenon with logarithmic nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper we highlight a type of hyperbolic equation relating the logarithmic source term with distributed delay and dynamic boundary condition. We get, under comfortable primary data is the weak solution to local existence. The results of the solutions were found using the Faydo–Galerkin method and Schoder’s fixed point theorem. Then, the minimum blow-up result was studied. Our work is an extension of some previous work. [ABSTRACT FROM AUTHOR]

In this present paper, we concern investigating nonlinear Kirchhoff-type problems subject to Dirichlet boundary conditions, incorporating nonlocal terms and logarithmic nonlinearity in the ϕ -Hilfer fractional spaces with the η (·) -Laplacian operator by means of the do Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities - Δ u + ϕ u = | u | p - 2 u log | u | 2 + λ f (u) + h (x) , in Ω , - Δ ϕ - ε 4 Δ 4 ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , where 4 < p < + ∞ , ε , λ > 0 are parameters, Δ 4 ϕ = div (| ∇ ϕ | 2 ∇ ϕ) , Ω ⊂ R 2 is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for λ large enough and ε fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters ε . [ABSTRACT FROM AUTHOR]

A nonlinear viscoelastic plate equation with logarithmic nonlinearity and variable-exponents is studied. Firstly, the global existence is showed. Next, by using an integral inequality due to Komornik the general decay result is obtained. Finally, the blow-up of solutions is proved with negative initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: − M ∫ Ω ∇ u p d x Δ p u = u p ∗ − 2 u + λ u p − 2 u − u p − 2 u ln u 2           x ∈ Ω ,                                                                         u = 0                                                                                   x ∈ ∂ Ω ,                                   where Ω ⊂ ℝ N is a bounded domain with a smooth boundary, 2 < p < p ∗ < N , and both p and N are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new method to prove the blowup at $ +\infty $ for the solution of the initial boundary value problem for a class of semilinear parabolic equations with logarithmic nonlinearity at sub-critical initial energy level by contradiction. Moreover, following the idea of contradiction, we obtained a sufficient criterion for the blowup at $ +\infty $ of the solution with arbitrarily positive initial energy by defining a new auxiliary function, in which the location of the initial data concerning the unstable set is not required. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the stabilization and blow up of the wave equation with infinite memory, logarithmic nonlinearity and acoustic boundary conditions. The authors discuss the existence of global solutions for the initial energy less than the depth of the potential well and investigate the energy decay estimates by introducing a Lyapunov function. Moreover, the authors establish the finite time blow up results of solutions and give the blow up time with upper bounded initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we considered a singular parabolic p -Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction. [ABSTRACT FROM AUTHOR]

This paper deals with the following (p,N)-Laplacian equation with logarithmic and critical exponential nonlinearities. Precisely, we study the problem ... where Ω ⊂ RN is a bounded domain, N ≥ 2, 1 < p < N < q, λ > 0 is a positive real parameter. By applying variational methods, we obtain the existence of solutions. [ABSTRACT FROM AUTHOR]

In the current paper, a generalized nonlinear Schrödinger (gNLS) equation with logarithmic nonlinearity is studied as a model for the propagation of optical pulses. More precisely, after applying a specific hypothesis for the solution of the governing equation, its Gaussian solitary wave is retrieved using the ansatz method. Some numerical simulations in two- and three-dimensional postures are presented to investigate the impact of different physical parameters on Gaussian solitary wave' dynamics. Results confirm that the physical parameters of the gNLS equation have a key role in controlling the dynamics of the Gaussian solitary wave. [ABSTRACT FROM AUTHOR]

This paper deals with the infinite blow‐up phenomena for a class of damped plate equations with logarithmic nonlinearity under the Navier boundary condition. Combining potential well method and modified differential inequality technique, we establish the infinite blow‐up result of solutions with arbitrary initial energy. In particular, it is not necessary to suppose that the initial velocity and the initial displacement should have the same sign in the sense of the L2${L^2}$ inner product, that is, the solution may blow up at infinity even ∫Ωu0u1dx<0$\int _\Omega {{u_0}{u_1}dx} < 0$, more precisely, ∫Ωu0u1dx>−12u0∗2$\int _\Omega {{u_0}{u_1}dx} > - \frac{1}{2}\left\Vert {{u_0}} \right\Vert _*^2$. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and asymptotic behavior for a singular epitaxial thin-film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut-off and combining with Hardy-Sobolev inequality, the technique of Faedo-Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and the lifespan are derived. [ABSTRACT FROM AUTHOR]

By means of variational methods, we study the existence and multiplicity of nontrivial weak solutions for a nonlocal τ(m)-Laplacian-like problem, arising from a capillarity phenomenon, with logarithmic nonlinearity and without the Ambrosetti-Rabinwitz condition, in the setting of variable exponents of Sobolev spaces in compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

This paper is concerned with the interaction between logarithmic source term and p-Laplacian term for nonlinear damped semilinear wave equation. We established the local existence and uniquenes under appropriate conditions. [ABSTRACT FROM AUTHOR]

A nonlinear Kirchhoff-type equation with logarithmic nonlinearity and variableexponents is studied. Firstly, the global existence is shown. Next, by using an integral inequality due to Komornik the general decay result is obtained. Finally, the blow-up of the solutions is proved. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: \[ \begin{cases} \displaystyle M\left(\int_{\mathbb R^{2N}}\frac{|u(x) - u(y)|^{\frac{N}{s}}}{|x - y|^{2N}}\,{\rm d}x\,{\rm d}y \right)(-\Delta)_{N/s}^s u \\ \quad = |u|^{q-2}u\ln |u|^2 + \mu f(u), & {\rm in}\ \Omega,\\ u=0, & {\rm in}\ \mathbb R^N\setminus \Omega, \end{cases} \] { M (∫ R 2 N | u (x) − u (y) | N s | x − y | 2 N d x d y) (− Δ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf (u) , in Ω , u = 0 , in R N ∖ Ω , where $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold. [ABSTRACT FROM AUTHOR]

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Considered herein is the well-posedness, asymptotic stability and blow-up of the initial-boundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity u tt - 1 x (x u x) x - 1 x (x u xt) x + ∫ 0 t m (t - λ) 1 x (x u x (x , λ)) x d λ = | u | r - 2 u ln | u | subject to a nonlocal boundary condition. Through the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differential-integral inequality techniques, we firstly demonstrate the global existence and uniqueness of weak solutions in certain weighted Sobolev spaces; Secondly, we establish the explicit polynomial and exponential energy decay estimates under some suitable conditions; Finally, we investigate the finite time blow-up criterion and then derive its upper and lower bounds of blow-up time. The above conclusions extend and improve some results in the literatures. [ABSTRACT FROM AUTHOR]

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u = b e − a x 2 , a < 0 , a , b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u = b e − a x 2 2 , respectively, for the third-order PDE, u = A e i Φ , where the amplitude b and the phase function a are complex-valued functions, A > 0 , and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota's approach or the method of simplest equation. [ABSTRACT FROM AUTHOR]

Higher-order statistics investigate the phase relationships between frequency components, an aspect which cannot be treated using conventional spectral measures such as the power spectrum. Among the widely used higher-order statistics, the bispectrum ranks prominently. By delving into higher-order correlations, the bispectrum offers a means of extracting additional merits and insights from frequency coupling, enhancing our understanding of complex signal interactions. This analytical approach overcomes the limitations of traditional methods, providing a more comprehensive view of the complex relationships within the frequency domain. In this paper, the extensive use of the bispectrum in various scientific and technical areas is firstly emphasized by presenting very recent applications. The main scope of this work is to investigate the consequences of various non-linearities in the creation of phase couplings. Specifically, the quadratic, the cubic and the logarithmic non-linearities are examined. In addition, simple recommendations are given on how the underlying nonlinearity could be detected. The total approach is novel, considering the capability to distinguish from the bispectral content if two non-linearities are simultaneously present. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following N-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities: - a + b ∫ Ω | ∇ u | N d x Δ N u = | u | p - 2 u ln | u | 2 + λ f (u) , in Ω , u = 0 , on ∂ Ω , where f(t) behaves like e x p α | t | N N - 1 . Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u b with precisely two nodal domains. Moreover, we show that the energy of u b is strictly larger than two times of the ground state energy and analyze the convergence property of u b as b ↘ 0 . [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem for a pseudo-parabolic type $ p $-Kirchhoff equation with logarithmic nonlinearity is investigated. By proving the invariance of the unstable set under the semi-flow of this problem and adopting the Levine's concavity argument, a general finite time blow-up criterion for this problem is established, which in particular implies that for some initial data, the problem admits finite time blow-up solutions at arbitrarily high initial energy level. Moreover, the lifespan of the blow-up solutions is estimated from above. [ABSTRACT FROM AUTHOR]

Abstract We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.

Abstract By using a direct non-Nehari manifold method from (Tang and Cheng in J. Differ. Equ. 261:2384–2402, 2016), we obtain an existence result of ground-state sign-changing homoclinic solutions that only changes sign once and ground-state homoclinic solutions for a class of discrete nonlinear p-Laplacian equations with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground-state energy is larger than twice the ground-state energy.

Abstract This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo–Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time.

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.