Large solution of a semilinear elliptic problem

AbstractWe show that large positive solutions exist for the following equationΔu+|∇u|q=p(x)f(u)(p+)in Ω ⊆ RN (N ≥ 3) in which the domain Ω is either bounded or equal to RN. The nonnegative function p is continuous and may vanish on large parts of Ω. If Ω = RN, then p must satisfy a decay condition∫0∞rϕ(r)dr<∞,whereϕ(r)=max⁡|x|=rp(x)as|x|→∞.Furthermore, we show that the given conditions on p are nearly optimal for equation (p+)

Propagation of Mild Singularities in Higher Dimensional Thermoelasticity

AbstractThe propagation of mild singularities for the semilinear model of three-dimensional thermoelasticity is studied. It is shown that the propagation picture of such singularities of the solution to the semilinear model coincides with one of the solutions to the corresponding linear model. As a simple consequence of our method, a similar result for the full semilinear Cauchy problem of one-dimensional thermoelasticity is also presented

Well-posedness of The Prandtl Equation in Sobolev Spaces

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow