Energy solutions of the Cauchy-Neumann problem for porous medium equations
Goro Akagi
Conference Publications 2009, 2009(Special): 1-10 doi: 10.3934/proc.2009.2009.1
The existence of energy solutions to the Cauchy-Neumann problem for the porous medium equation of the form $v_t - \Delta (|v|^{m-2}v) = \alpha v$ with $m \geq 2$ and $\alpha \in \mathbb{R}$ is proved, by reducing the equation to an evolution equation involving two subdifferential operators and exploiting subdifferential calculus recently developed by the author.
keywords: subdifferential operator Neumann boundary condition doubly nonlinear evolution equation Porous medium equation reflexive Banach space
Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity
Goro Akagi
Conference Publications 2005, 2005(Special): 30-39 doi: 10.3934/proc.2005.2005.30
This paper is intended as an investigation of the solvability of Cauchy problem for doubly nonlinear evolution equation of the form $dv(t)/dt + \partial \lambda^t(u(t)) \in 3 f(t)$, $v(t) \in \partial \psi(u(t))$, 0 < $t$ < $T$, where $\partial \lambda^t$ and $\partial \psi$ are subdifferential operators, and @'t depends on t explicitly. Our method of proof relies on chain rules for t-dependent subdifferentials and an appropriate boundedness condition on $\partial \lambda^t$ however, it does not require either a strong monotonicity condition or a boundedness condition on $\partial \psi$. Moreover, an initial-boundary value problem for a nonlinear parabolic equation arising from an approximation of Bean's critical-state model for type-II superconductivity is also treated as an application of our abstract theory.
keywords: Doubly nonlinear evolution equation reflexive Banach space time-dependent subdifferential Bean model.
Doubly nonlinear parabolic equations involving variable exponents
Goro Akagi
Discrete & Continuous Dynamical Systems - S 2014, 7(1): 1-16 doi: 10.3934/dcdss.2014.7.1
This paper is concerned with doubly nonlinear parabolic equations involving variable exponents. The existence of solutions is proved by developing an abstract theory on doubly nonlinear evolution equations governed by gradient operators. In contrast to constant exponent cases, two nonlinear terms have inhomogeneous growth and some difficulty may occur in establishing energy estimates. Our method of proof relies on an efficient use of Legendre-Fenchel transforms of convex functionals and an energy method.
keywords: subdifferential parabolic variable exponent $p(x)$-Laplacian evolution equation. Doubly nonlinear
Evolution equations and subdifferentials in Banach spaces
Goro Akagi Mitsuharu Ôtani
Conference Publications 2003, 2003(Special): 11-20 doi: 10.3934/proc.2003.2003.11
Sufficient conditions for the existence of strong solutions to the Cauchy problem are given for the evolution equation $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t)) \in f(t)$ in V*, where $\partial\upsilon^1$ is the so-called subdifferential operator from a Banach space V into its dual space V* ($i$ = 1,2).
   Studies for this equation in the Hilbert space framework has been done by several authors. However the study in the V -V* setting is not pursued yet.
   Our method of proof relies on some approximation arguments in a Hilbert space. To carry out this procedure, it is assumed that there exists a Hilbert space H satisfying $V \subset H \-= H$* $\subset V$* with densely defined continuous injections.
   As an application of our abstract theory, the initial-boundary value problem is discussed for the nonlinear heat equation: $ut(x, t)-\Delta_p u(x, t)-|u|^(q-2) u(x, t) = f(x, t), x \in \Omega, u|_(\partial\Omega) = 0, t \>= 0$, where ­$\Omega$ is a bounded domain in $\mathbb(R)^N$. In particular, the local existence of solutions is assured under the so-called subcritical condition, i.e., $q < p$*, where $p$* denotes Sobolev’s critical exponent, provided that the initial data $u_0$ belongs to $W_0^(1,p)(\­Omega)$.
keywords: subdifferential global solution Sobolev's critical exponent local solution nonlinear heat equation subcritical. reflexive Banach space Evolution equation p-Laplacian
On a certain degenerate parabolic equation associated with the infinity-laplacian
Goro Akagi Kazumasa Suzuki
Conference Publications 2007, 2007(Special): 18-27 doi: 10.3934/proc.2007.2007.18
The comparison, uniqueness and existence of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate parabolic equation of the form $u_t$ = $\Delta_(\infty)u$, where $\Delta_(\infty)$ denotes the so-called infinity-Laplacian given by $\Delta_(\infty)u$ = $\Sigma^(N)_(i,j=1) u_x_i u_x_j u_(x_i)_x_j$ . Our proof relies on a coercive regularization of the equation, barrier function arguments and the stability of viscosity solutions.
keywords: viscosity solution. infinity-Laplacian Degenerate parabolic equation
Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian
Goro Akagi Kei Matsuura
Conference Publications 2011, 2011(Special): 22-31 doi: 10.3934/proc.2011.2011.22
This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation involving the so-called $p(x)$-Laplacian. A subdifferential approach is employed to obtain a well-posedness result as well as to investigate large-time behaviors of solutions.
keywords: $p(x)$-Laplacian subdifferential parabolic equation variable exponent Lebesgue and Sobolev spaces
Principle of symmetric criticality and evolution equations
Goro Akagi Jun Kobayashi Mitsuharu Ôtani
Conference Publications 2003, 2003(Special): 1-10 doi: 10.3934/proc.2003.2003.1
   Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais [6] gave some sufficient conditions to guarantee the so-called "Principle of Symmetric Criticality": every critical point of J restricted on the subspace of symmetric points becomes also a critical point of J on the whole space X. In [5], this principle was generalized to the case where J is non-smooth and the setting does not require the full variational structure when G is compact or isometric.
   The purpose of this paper is to combine this result with the abstract theory developed in [1] and [2] concerning the evolution equation: $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t))$ ∋ $f(t)$ in V*, where $\partial\upsilon^i$ is the so-called subdifferential operator from a Banach space X into its dual V*. It is assumed that there exists a Hilbert space H satisfying $V \subset H \subset V $ and that G acts on these spaces as isometries. In this setting, the existence of G-symmetric solution for above equation can be discussed.
   As an application, a parabolic problem with the p-Laplacian in unbounded domains is discussed.
keywords: symmetric criticality group action unbounded domain. evolution equations p-Laplacian subdifferential

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