We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator ∂t + ∆ 2 is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We ...

We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. prove that if-and-only-if relations between coefficients and derivatives of Dirichlet-to-Neumann map hold. Based on strong relations, we study a constructive global uniqueness inclusion detection Calderón inverse problem. Meanwhile, can also derive Lipschitz stability finitely man...

We consider the inverse problem of determining a potential in semilinear elliptic equation from knowledge Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that is uniquely determined by map measured at single boundary point, or integrated against fixed measure. This result valid even when Dirichlet data only given on small subset boundary. also give related uniqueness results Ri...

The numerical computation of solitary waves to semilinear elliptic equations in innnite cylinders is investigated. Rather than solving on the innnite cylinder, the equation is approximated by a boundary-value problem on a nite cylinder. Convergence and stability results for this algorithm are given. In addition, it is shown that Galerkin approximations can be used to calculate solitary waves fo...

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point met...

In this paper, we study the asymptotic behavior of positive solutions and apply the “improved moving plane” method to prove the symmetry of positive solutions of semilinear elliptic systems in unbounded cylinders.

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form 8><>>>: −∆u = g(x, v) , in Ω −∆v = f(x, u) , in Ω u > 0 , v > 0 in Ω

We obtain precise global bifurcation diagrams for both one-sign and sign-changing solutions of a semilinear elliptic equation, for the nonlinearity being asymptotically linear. Our method combines the bifurcation approach and spectral analysis.

We prove the exact multiplicity of positive boundary blow-up solutions to a semilinear elliptic equation with bistable nonlinearity for the one-dimensional case. We use time-mapping techniques to determine the exact shape of the bifurcation diagram.

We consider the asymptotic analysis and some existence result on blowing up solutions for a semilinear elliptic equation in dimension 2 with nonlinear exponential term, singular sources and Dirichlet boundary condition.