In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory.

In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

We establish the interior Holder continuity for locally bounded solutions, and the Harnack inequality for non-negative continuous solutions to a class of anisotropic elliptic equations with bounded and measurable coefficients, whose prototype equation is u(xx) + Delta(q,y) u = 0 locally in R x RN-1, for q < 2, via ideas and tools originating from the regularity theory for degenerate and singular parabolic equations.

We prove Harnack type inequalities for a wide class of parabolic doubly nonlinear equations including $u_t=$ ${ div}(|u|^{m-1}|Du|^{p-2}Du)$. We will distinguish between the supercritical range $3 - \frac {p} {N} < p+m < 3$ and the subcritical $2 < p+m \le 3 - \frac {p} {N}$ range. Our results extend similar estimates holding for general equations having the same structure as the parabolic $p$-Laplace or the porous medium equation and recently collected in [6].

We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is $$\begin{aligned} \partial _t \big (|u|^{\alpha -1}u \big ) - \sum ^N_{i=1} \partial _i \big ( |\partial _i u|^{p_i - 2} \partial _i u \big ) = 0, \end{aligned}$$ ∂ t ( | u | α - 1 u ) - ∑ i = 1 N ∂ i ( | ∂ i u | p i - 2 ∂ i u ) = 0 , where $$\alpha > 0$$ α > 0 and $$p_i \in (1, \infty )$$ p i ∈ ( 1 , ∞ ) . We obtain super and ultracontractive bounds, and global boundedness in space for solutions to the Cauchy problem with initial data in $$L^{\alpha +1}(\mathbb {R}^N)$$ L α + 1 ( R N ) , and show that the mass is nonincreasing over time. As a consequence, compactly supported evolution is shown for optimal exponents. We introduce a seemingly new paradigm, by showing that Caccioppoli estimates, local boundedness and semicontinuity are consequences of the membership to a suitable energy class. This membership is proved by first establishing the continuity of the map $$t \mapsto |u|^{\alpha -1}u(\cdot ,t) \in L^{1+1/\alpha }_{\text {loc}}(\Omega )$$ t ↦ | u | α - 1 u ( · , t ) ∈ L loc 1 + 1 / α ( Ω ) permitting us to use a suitable mollified weak formulation along with an appropriate test function.