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With respect to generalized trigonometric functions, since the discovery of double-angle formula for a special case by Edmunds, Gurka and Lang in 2012, no double-angle formulas have been found. In this paper, we will establish new double-angle formulas of generalized trigonometric functions in two special cases., Comment: 7 pages

The Cauchy problem for a parabolic partial differential equation with a power nonlinearity is studied. It is known that in some parameter range, there exists a time-local solution whose singularity has the same asymptotics as that of a singular steady state. In this paper, a sufficient condition for initial data is given for the existence of a solution with a moving singularity that becomes anomalous in finite time.

We investigate the initial–boundary value problem{ut=Δu+vpin Ω×(0,T),vt=Δv+uqin Ω×(0,T),u(x,t)=v(x,t)=0on ∂Ω×(0,T),u(x,0)=ρφ(x),v(x,0)=ρψ(x)in Ω, where p,q⩾1 and pq>1, Ω is a bounded domain in RN with a smooth boundary ∂Ω, ρ>0 is a parameter, φ(x) and ψ(x) are nonnegative continuous functions on Ω¯. We show that the life span (or blow-up time) of the solution of this problem approaches the life span of the solution of the ODE system obtained when dropping the diffusion terms as ρ→∞. The proof is based on the comparison principle and Kaplanʼs method.

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a solution with a moving singularity that becomes anomalous in finite time. Our concern is a blow-up solution with a moving singularity. In this paper, we show that there exists a solution with a moving singularity such that it blows up at space infinity.

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux–Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.

We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.

We study the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a time-local solution with prescribed moving singularities. Our concern in this paper is the existence of a time-global solution. By using a perturbed Haraux-Weissler equation, it is shown that there exists a forward self-similar solution with a moving singularity. Using this result, we also obtain a sufficient condition for the global existence of solutions with a moving singularity.

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.

We investigate the initial-boundary problem { u t = Δ u + f ( u ) in Ω × ( 0 , ∞ ) , u = 0 on ∂ Ω × ( 0 , ∞ ) , u ( x , 0 ) = ρ φ ( x ) in Ω , where Ω is a bounded domain in R N with a smooth boundary ∂Ω, ρ > 0 , φ ( x ) is a nonnegative continuous function on Ω ¯ , f ( u ) is a nonnegative superlinear continuous function on [ 0 , ∞ ) . We show that the life span (or blow-up time) of the solution of this problem, denoted by T ( ρ ) , satisfies T ( ρ ) = ∫ ρ ‖ φ ‖ ∞ ∞ d u f ( u ) + h.o.t. as ρ → ∞ . Moreover, when the maximum of φ is attained at a finite number of points in Ω, we can determine the higher-order term of T ( ρ ) which depends on the minimal value of | Δ φ | at the maximal points of φ. The proof is based on a careful construction of a supersolution and a subsolution.