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### Open Access Journals

PROC

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.

PROC

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.

DCDS-S

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

PROC

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 $W_0^(1,p)(\Omega)$.

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 $W_0^(1,p)(\Omega)$.

PROC

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.

PROC

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.

PROC

Let

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

As an application, a parabolic problem with the p-Laplacian in unbounded domains is discussed.

**be a Banach space on which a symmetry group***X***linearly acts and let***G***be a***J***-invariant functional defined on***G***. In 1979, R. Palais [6] gave some sufficient conditions to guarantee the so-called "Principle of Symmetric Criticality": every critical point of***X***restricted on the subspace of symmetric points becomes also a critical point of***J***on the whole space***J***. In [5], this principle was generalized to the case where***X***is non-smooth and the setting does not require the full variational structure when***J***is compact or isometric.***G*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

***, where $\partial\upsilon^i$ is the so-called subdifferential operator from a Banach space***V***into its dual***X****. It is assumed that there exists a Hilbert space***V***satisfying $V \subset H \subset V $ and that***H***acts on these spaces as isometries. In this setting, the existence of***G***-symmetric solution for above equation can be discussed.***G*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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