• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1597-1612
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• 2016

There are fruitful results on degenerate parabolic-hyperbolic equations recently following the idea of $Kru{\check{z}}kov^{\prime}s$ doubling variables device. This paper is devoted to the well-posedness of nonhomogeneous boundary problem for degenerate parabolic-hyperbolic equations with spatially dependent second order operator, which has not caused much attention. The novelty is that we use the boundary flux triple instead of boundary layer to treat this problem.

• Korean Journal of Mathematics
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• v.5 no.2
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• pp.151-167
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• 1997

We estimate the interior Lipschitz norm and maximum of the solution for degenerate parabolic equations with absorption. Also obtain the growth rate of the solution $u$ in terms of time $t$. From this we show the uniqueness of solution with respect to the initial trace.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.751-766
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• 2013

We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.423-443
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• 2022

We study the existence and long-time behavior of weak solutions to a class of strongly degenerate semilinear parabolic equations with exponential nonlinearities on ℝ^N. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of a global attractor is proved by combining the tail estimates method and the asymptotic a priori estimate method.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.303-316
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• 2000

The Cauchy problem for degenerate parabolic equations with absorption is studied. We prove the existence of a fundamental solution. Also a Harnack type inequality is established and the existence and uniqueness of initial trace for nonnegative solutions is shown.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.915-927
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• 2000

This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.447-463
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• 2021

In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities. First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces. Then, we prove the existence of global attractors by using the asymptotic a priori estimates method.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.145-161
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• 2020

In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.

• Journal for History of Mathematics
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• v.24 no.1
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• pp.31-44
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• 2011

Partial differential equations on a Riemannain manifold is one of the most important areas in differential geometry. In this article, we survey the role of parabolic equations on some of the main results of differential geometry and topology, especially in the modern mathematical history. Also, we introduce some recent research in this area.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.91-106
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• 2004

We prove the regularity of a moving interface of the solutions of the initial value problem of equation (1.1) is $C^{\infty}$.