Parabolic problems with nonlinear boundary conditions and critical nonlinearities

@article{Arrieta1999ParabolicPW,
  title={Parabolic problems with nonlinear boundary conditions and critical nonlinearities},
  author={Jos{\'e} M. Arrieta and Alexandre Nolasco de Carvalho and An{\'i}bal Rodr{\'i}guez-Bernal},
  journal={Journal of Differential Equations},
  year={1999},
  volume={156},
  pages={376-406}
}
We prove existence, uniqueness and regularity of solutions For heat equations with nonlinear boundary conditions. We study these problems with initial data in L-q(Omega), W-1,W-q(Omega), 1 < q < infinity or measures and with critically growing non-linearities. 

Figures from this paper

Some Qualitative Dynamics of Nonlinear Boundary Conditions
TLDR
The results range from local existence and regularity of solutions, to global existence, dissipativeness and existence of attractors, and to blow-up in finite time.
Existence and multiplicity of self-similar solutions for heat equations with nonlinear boundary conditions
This paper is concerned with self-similar solutions in the half-space for linear and semilinear heat equations with nonlinear boundary conditions. Existence, multiplicity and positivity of these
Spatial Homogeneity in Parabolic Problems With Nonlinear Boundary Conditions
In this work we prove that global attractors of systems of weakly coupled parabolic equations with nonlinear boundary conditions and large diffusivity are close to attractors of an ordinary
Solvability of the Heat Equation with a Nonlinear Boundary Condition
TLDR
The relationship between the life span of the solution and the behavior of the initial function is studied and the solvability of the heat equation in a half-space of N with a nonlinear boundary condition is obtained.
Nonlinear boundary value problem for pseudoparabolic equation
Abstract An initial boundary value problem for the linear pseudoparabolic equation ( u + η M u ) t + k ( t ) M u = f is considered under the nonlinear mixed boundary condition. M is a linear
Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces
We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we
Solvability of the First Boundary-Value Problem for the Heat-Conduction Equation with Nonlinear Sources and Strong Power Singularities
By using the Schauder principle and the principle of contracting mappings, we study the character of point power singularities for the solution of the generalized first boundary-value problem for the
Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary
Abstract We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this
Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary
We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this
Complete and energy blow-up in parabolic problems with nonlinear boundary conditions
We study the possible continuation of solutions of a nonlinear parabolic problem after the blow-up time. The nonlinearity in the equation is dissipative and blow-up is caused by the nonlinear
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 22 REFERENCES
Geometric Theory of Semilinear Parabolic Equations
Preliminaries.- Examples of nonlinear parabolic equations in physical, biological and engineering problems.- Existence, uniqueness and continuous dependence.- Dynamical systems and liapunov
Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations
PARABOLIC PROBLEMS WITH CRITICAL NONLINEARITIES AND APPLICATIONS TO NAVIER-STOKES AND HEAT EQUATIONS JOSÉ M. ARRIETA AND ALEXANDRE N. CARVALHO Abstract. We prove a local existence and uniqueness
Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp
Abstract The equation Lu = ƒ;(x, u) on B × (0, ∞), B bounded, smooth domain in R n with nonlinear boundary conditions ∂u ∂v = g(x, u) on ∂B × (0, ∞) is studied, L being the uniformly parabolic
Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems
It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary
Nonlinear Parabolic Equations Involving Measures as Initial Conditions
Abstract : The Cauchy problem is considered for certain equations with a boundary condition and an initial condition. A solution of the equations exists if and only if O p n+2/n. This paper deals
PARTIAL DIFFERENTIAL EQUATIONS
Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear
Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data
In this paper the authors consider a specified Cauchy problem for semilinear hear equations on [Re][sup n] and also the Cauchy problem for the Navier-Stokes equation on [Re][sup n] for n[ge]2 of a
Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations
Publisher Summary This chapter discusses the monotonicity methods in Hilbert spaces and presents some applications to nonlinear partial differential equations. It describes classical properties of
Semilinear evolution equations in Banach spaces
Abstract We study the evolution equation u′(t) = Au(t) + J(u(t)), t ⩾ 0, where etA is a C0 semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In
Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory
Parabolic Evolution Equations and their Applications This book details the mathematical developments in total variation based image restauration. From the reviews: "This book is devoted to PDE's of
...
1
2
3
...