Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p $ in $R^N $ with $p > 1$ and nonnegative initial values. Fujita showed that if $1 1 + {2 / N}$, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrodinger equation.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1048&context=math_pubs

The role of critical exponents in blowup theorems

Progress in Nonlinear Differential Equations and their Applications

https://ddd.uab.cat/pub/prepub/2009/hdl_2072_39459/Pr858.pdf

Positive solutions of nonlinear problems involving the square root of the Laplacian

The Role of Critical Exponents in Blow-Up Theorems: The Sequel

Unspecified Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22758 Originally published at: Chipot, M; Fila, M; Quittner, P (1991). Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Mathematica Universitatis Comenianae. New Series, 60(1):35-103. Acta Math. Univ. Comenianae Vol. LX, 1(1991), pp. 35–103 35 STATIONARY SOLUTIONS, BLOW UP AND CONVERGENCE TO STATIONARY SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS M. CHIPOT, M. FILA AND P. QUITTNER

Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions

In this paper positive solutions of the heat equation with a nonlinear Neumann boundary conditions in an upper halfspace are studied. The optimal result on blow-up rate, valid for all solutions which blow up in finite time, is established under the assumption that the exponent of a nonlinear boundary condition is subcritical in the Sobolev sense. This complements results derived for the bounded domain case in [10, 13] either for monotonous solutions or under a stronger restriction on the exponent of a boundary condition.

On the blow-up rate for the heat equation with a nonlinear boundary condition

We describe all nontrivial nonnegative solutions to the problem $-\Delta u=au^{(n+2)/(n-2)}$ in $H,\ \partial u/\partial\nu=bu^{n/(n-2)}$ on $\partial H$, where $H$ is a half-space of ${\bf R}^n\ (n\geq3)$.

/pdf/on-the-solutions-to-some-elliptic-equations-with-nonlinear-1wroocnczp.pdf

On the solutions to some elliptic equations with nonlinear Neumann boundary conditions

Existence of Positive Solutions of a Semilinear Elliptic Equation in Rn + with a Nonlinear Boundary Condition

In this paper, we consider the system u t = u, v t = v x 2R N , t > 0, @u @x 1 = v p , @v @x 1 = u q x 1 = 0, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x) x2R N , where R N = {(x 1 ,x 0 ) | x 0 2 R N 1 ,x 1 > 0}, p,q > 0, and u 0 , v 0 nonnegative. We prove that if pq 1 every nonnegative solution is global. When pq > 1 we let = 1 p+1 pq 1 , = 1 2 q+1 pq 1 . We show that if max(,) N 2 , all nontrivial nonnegative solutions are nonglobal; whereas if max(,) 0} (N 1), p,q > 0, and both u 0 (x) and v 0 (x) are nonnegative bounded functions satisfying the compatibility condition

/pdf/on-critical-exponents-for-a-system-of-heat-equations-coupled-2w3eofiby6.pdf

Marek Fila

Papers

Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions

On the blow-up rate for the heat equation with a nonlinear boundary condition

On the solutions to some elliptic equations with nonlinear Neumann boundary conditions

Existence of Positive Solutions of a Semilinear Elliptic Equation in Rn + with a Nonlinear Boundary Condition

On critical exponents for a system of heat equations coupled in the boundary conditions