Spectrum of some integro-differential operators and stability of travelling waves
The authors study some spectral properties of the linear integro-differential operator \[ Lu=u''+cu'+a(x)\phi\ast u+b(x)u \] acting from the Banach space \(C_0^2(\mathbb{R})\) into the Banach space \(C_0(\mathbb{R})\), where \(C_0(\mathbb{R})\) consists of all continuous functions \(u\) with \(u(\pm \infty)=0\), \(C_0^2(\mathbb{R})\) consists of all \(C^2\) functions \(u\) with \(u,u',u''\in C_0^2(\mathbb{R})\), \(a(x)\) and \(b(x)\) are both bounded and continous functions that have limits \(a^\pm\) and \(b^\pm\) when \(x\to \pm \infty\), and \(\phi\) is some integrable function. The main result shows that the principal eigenvalue of \(L\) is characterized by the positiveness of the corresponding eigenfunction under appropriate assumptions. In particular, if \(Lu_0=0 \) with \(u_0>0\) and further, if \(a\geq 0\) and \(a^\pm+b^\pm <0\), then \(0\) is a simple eigenvalue, while all other eigenvalues have negative real part. This result plays an important role in the study of stability of traveling waves of reaction-diffusion equations with nonlocal interaction such as \[ \partial_t u=\partial_{xx}u+f(u,\phi\ast u), \] the linearization of which around a traveling wave \(w(x-ct)\) has the form \(\partial_t u(t,x)=L[u(t,\cdot)](x)\) with \(a(x)=\partial_2 f (w(x),\phi\ast w(x))\) and \(b(x)=\partial_1 f (w(x),\phi\ast w(x))\). By this result, the authors further provide the minimax representation of the wave speed. Some examples of applications are illustrated, and it is also pointed out that the result applies to time-delayed reaction diffusion models.
scientific article; zbMATH DE number 5915043
scientific article