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This book provides a comprehensive analysis of the long-term behavior of three important classes of nonlinear partial differential equations with complex structures. It investigates viscoelastic Petrovsky equations with strong damping and time-varying delay, Laplacian equations with variable exponent sources, and reaction-diffusion equations featuring singular coefficients and variable exponent nonlinearities. The work focuses on fundamental phenomena including finite-time blow-up, global existence, and energy decay. Key achievements include establishing sharp conditions for blow-up with explicit upper and lower time estimates, proving general decay rates beyond standard exponential or polynomial decay, demonstrating well-posedness under non-standard conditions, and providing complete classifications of solution behavior across different energy regimes. The book serves as an advanced reference for researchers in applied mathematics and mathematical physics, offering new analytical frameworks for systems with memory effects, variable exponents, singular coefficients, and delay mechanisms.
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