# 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.

## 116 Citations

Some Qualitative Dynamics of Nonlinear Boundary Conditions

- Mathematics, Computer ScienceInt. J. Bifurc. Chaos
- 2002

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

- Mathematics
- 2015

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

- Mathematics
- 2004

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

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2019

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

- Mathematics
- 2021

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

- Physics, Mathematics
- 2014

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

- Mathematics
- 2014

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

- Mathematics
- 2011

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

- Mathematics
- 2011

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

- Mathematics
- 2005

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…

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