arXiv subjects

Tae Gab Ha

Publications and source records attributed to Tae Gab Ha.

6 recordsLinked to original sources

arXiv2026

Global existence and finite-time blow-up for a strongly damped wave--MGT system with fully subcritical logarithmic nonlinearity

We study a coupled strongly damped wave--Moore--Gibson--Thompson (MGT) system with logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$ in the full Sobolev-subcritical range $2<\gamma<2^*:=2n/(n-2)$. The main difficulty is that, in this full range, the logarithmic nonlinearity does not admit the standard compactness and difference estimates available in the lower subcritical regime. Using the augmented variable $w=v+\tau v_t$, we derive the exact energy-dissipation identity for the strongly damped system and exploit the associated coupled potential-well structure. For the local theory, a Faedo--Galerkin scheme combined with a closed nonlinear differential inequality and a spatial domain-splitting argument yields local existence. The strong damping provides the additional regularity needed to prove uniqueness throughout the full subcritical range and to establish a continuation principle. Below the well depth $d_\alpha$, we then obtain a sharp dynamical dichotomy: solutions with initial data in the stable set exist globally and decay exponentially, whereas solutions with initial data in the unstable set blow up in finite time. The blow-up argument reveals a structural cancellation specific to the augmented formulation: once the logarithmic contribution is reconstructed through the exact energy identity, the unfavorable MGT residual in Levine's concavity functional is absorbed algebraically. This allows the concavity method to close without imposing any additional sign condition on the initial velocities.

math.AP↗

arXiv2026

Global Well-Posedness and Conditional Asymptotic Stability for a Coupled Wave-MGT System with Logarithmic Nonlinearity

We study a coupled system formed by a conservative wave equation and a dissipative Moore-Gibson-Thompson (MGT) equation on a bounded domain. The wave component is driven by the logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$, $2<\gamma<\frac{2(n-1)}{n-2}$, and carries no direct damping. Rather than employing cross-multiplier arguments, we introduce the coupled variable $w=v+\tau v_{t}$, which reveals the exact energy structure associated with the interaction term. This formulation yields a genuine coupled energy together with a coercive quadratic form $Q_{\alpha}(u,w)=\norm{\nabla u}_{2}^{2}+\norm{\nabla w}_{2}^{2}+2\alpha(u,w)$, provided that $|\alpha|<\lambda_{1}$. Based on this structure, we construct a coupled potential well and prove global well-posedness of weak solutions for initial data lying below the corresponding well depth and inside the stable set. We also show that the energy is strictly dissipative through the MGT component. In addition, a modal analysis of the linearized system identifies a high frequency spectral obstruction to uniform exponential decay, quantifying the weakness of the dissipation transfer to the wave branch. Finally, assuming the relative compactness of the trajectory in the natural energy space and imposing $0<|\alpha|<\lambda_1$, we apply LaSalle's invariance principle to establish conditional asymptotic stability of the zero equilibrium. The result provides a structurally consistent indirect stabilization theorem for the coupled wave--MGT dynamics without relying on unjustified exponential decay claims.

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arXiv2026

Stabilization for the wave equation with fully subciritical logarithmic nonlinearity

In this paper, we consider a wave equation with strong damping and logarithmic nonlinearity. This paper aims to study the local and global existence, uniqueness and the uniform energy decay rate of a weak solution under some sufficient conditions on the initial data. Unlike previous literature restricted to the lower subcritical range $2 < \gamma < \frac{2(n-1)}{n-2}$, we successfully extend the validity of the well-posedness and stabilization results to the upper subcritical range $\frac{2(n-1)}{n-2} \leq \gamma < \frac{2n}{n-2}$.

math.AP↗

arXiv2024

Global solutions and blow-up for the wave equation with variable coefficients: II. boundary supercritical source

In this paper, we consider the wave equation with variable coefficients and boundary damping and supercritical source terms. The goal of this work is devoted to prove the local and global existence, and classify decay rate of energy depending on the growth near zero on the damping term. Moreover, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy.

math.AP↗

arXiv2023

Finite time blow-up in higher dimensional two species problem in the Cauchy problem

In this paper, we study the blow-up radial solution of fully parabolic system with higher dimensional two species Cauchy problem for some initial condition. In addition, we show that the set of positive radial functions in $L^{1}(\mathbb{R})\cap BUC(\mathbb{R}^{N}) \times L^{1}(\mathbb{R})\cap BUC(\mathbb{R}^{N}) \times W^{1,1}(\mathbb{R}^{N}) \cap W^{1,\infty}(\mathbb{R}^{N})$ has a dense subset composed of positive radial initial data causing blow-up in finite time with respect to topology $L^{p}(\mathbb{R}^{N}) \times L^{p}(\mathbb{R}^{N}) \times H^{1}(\mathbb{R}^{N})\cap W^{1,1}(\mathbb{R}^{N})$ for $p \in \left[1,\frac{2N}{N+2}\right)$.

math.AP↗

arXiv2023

Global bounded solution of the chemotaxis attraction repulsion Cauchy problem with the nonlinear signal production in $\mathbb{R}^{N}$

In this paper, we consider the following attraction repulsion chemotaxis model with nonlinear signal term: \begin{align*} &u_{t}=\nabla \cdot(\nabla u-ξ_{1} u \nabla v +ξ_{2} u \nabla w), \quad &0=Δv -λ_{1}v +f_{1}(u), \quad &0=Δw -λ_{2}w +f_{2}(u), \quad x \in \mathbb{R}^{N}, t>0, \end{align*} where $ξ_{1},ξ_{2},λ_{1},λ_{2}$ are for some positive constants, and \begin{equation*} f_{1} \in C^{1}([0,\infty)) \; \text{satisfying} \; 0 \leqslant f_{1}(s) \leqslant c_{1}s^{l}, \; \forall s \geqslant 0 \ \text{and} \ l> 0, \end{equation*} \begin{equation*} f_{2} \in C^{1}([0,\infty))\; \text{satisfying} \; 0 \leqslant f_{2}(s) \leqslant c_{2}s^{m}, \; \forall s \geqslant 0 \ \text{and} \ m> 0. \end{equation*} We will show that this problem has a unique global bounded solution when $ l>\frac{2}{N}, l<m \ \text{with} \ m \geqslant 1$, or $l=m<\frac{2}{N}$.

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