Nonlinear Evolution Equations

This book investigates the blow-up phenomena, asymptotic behavior, and stability ofsolutions for several classes of nonlinear partial differential equations (PDEs), includingreaction-diffusion and wave-type equations with variable exponents, memory effects, andsingular coeffcients. The work is divided into four main parts.First, we study the blow-up phenomenon for nondegenerate parabolic PDEs in boundeddomains. By considering a nonnegative diffusion coeffcient a(x, t), we establish new blowup criteria and derive sharp lower and upper bounds for the blow-up time of semilinearreaction-diffusion equations and nonlinear equations involving the m(x, t)-Laplacian operator.Second, we analyze the initial-boundary value problem for Kirchhoff-type viscoelasticwave equations with Balakrishnan-Taylor damping, infinite memory, and time-varyingdelay. Under suitable assumptions on the relaxation function and initial data, we provethat the energy decays at a rate determined by the relaxation function, which may beneither exponential nor polynomial. Moreover, we establish a general stability resultunder a weak growth condition on the relaxation kernel.