Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Abstract.

Our first basic model is the fully nonlinear dual porous medium equation with source

$$u_{t} = {\left| {\Delta u} \right|}^{{m - 1}} \Delta u + u^{p} \quad in\;{\mathbb{R}}^{N} \times {\mathbb{R}}_{ + } ,\quad m > 1,\;\; p > 1,$$

for which we consider the Cauchy problem with given nonnegative bounded initial data u ₀. For the semilinear case m=1, the critical exponent $p_{0} = 1 + \frac{2}{N}$ was obtained by H. Fujita in 1966. For p ∈(1, p ₀] any nontrivial solution blows up in finite time, while for p > p ₀ there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations with differential operators in divergent form, where classical techniques associated with weak solutions and integration by parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches to calculating critical Fujita exponents in different functional settings.

The second models with a “semi-divergent” diffusion operator is the thin film equation with source

$$u_{t} = - \nabla \cdot (u^{n} \nabla \Delta u) + u^{p} \quad in\;{\mathbb{R}}^{N} \times {\mathbb{R}}_{ + } ,\quad n > 0,\;\;p > n + 1\quad (u \geq 0),$$

for which the critical exponent is shown to be $p_{0} = n + 1 + \frac{4}{N}.$