arXiv subjects

Soyeun Jung

Publications and source records attributed to Soyeun Jung.

11 recordsLinked to original sources

arXiv2026

Traveling waves for combustion reaction-diffusion-convection equations: the full range of wave speeds

We consider traveling wave solutions to a reaction--diffusion--convection equation with a combustion-type reaction term. While a necessary condition for the existence of traveling waves is $c \geq H^*:=\sup_{0<u \leq θ} \big(-\frac{1}{u} \int_0^u h(σ)\,dσ\big)$, where $c$ denotes the wave speed, $θ\in(0,1)$ the ignition threshold, and $h$ the convective term, the available results in \cite{DZ25,MM03} establish existence and nonexistence only under the restriction $c \geq -\min_{u\in[0,1]} h(u)$. In this note, we close this gap by covering the entire range $c\ge H^*$.

math.AP↗

arXiv2026

Traveling waves for bistable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

Continuing our previous study \cite{DJKZ} on the monostable reaction-diffusion-convection equation, we analyze the bistable case under weak regularity assumptions. Our approach applies monostable results on the subintervals where the reaction term $g$ has constant sign, thereby establishing both existence and nonexistence of bistable traveling wave solutions. We extend the results of \cite{MMM04}, obtained for $p=2$ under higher regularity assumptions ($d \in C^1[0,1]$, $g,h \in C[0,1]$), to the $p$-Laplacian with $p>1$ in our weak regularity setting.

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arXiv2026

Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms.

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arXiv2025

Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

Within the framework developed in \cite{Gr, JLL, RT1}, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter $σ$ ranges over an unbounded domain.

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arXiv2020

Stability of strong detonation waves for Majda's model with general ignition functions

For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.

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arXiv2017

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.

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arXiv2016

Diffusive stability of spatially periodic solutions of the Brusselator model

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg Landau amplitude equations, rigorously validating the standard weakly unstable approximation and Eckhaus criterion.

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arXiv2016

Pointwise nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves

In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically.

math.AP↗

arXiv2012

Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws

In the previous paper \cite{J1}, we established pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves $\bar u$ of a system of reaction diffusion equations, and also obtained pointwise nonlinear stability and behavior of $\bar u$ under small perturbations. In this paper, using periodic resolvent kernels and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with periodic standing waves $\bar u$ of a system of conservation laws. We also show pointwise nonlinear stability of $\bar u$ by estimating decay of modulated perturbation $v$ of $\bar u$ under small perturbation $|v_0| \leq E_0(1+|x|)^{-3/2}$ for sufficiently small $E_0>0$.

math.AP↗

arXiv2012

Stability of ZND detonations for Majda's model

We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda's model for reacting flow, and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of RoqueJoffre-Vila that viscous detonations of Majda's model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.

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arXiv2011

Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves

By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations.With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we obtain $L^p$- behavior($p \geq 1$) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in $L^1 \cap H^1$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|\leq E_0e^{-|x|^2/M}$ and $|u_0| \leq E_0(1+|x|)^{-3/2}$, respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.

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