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This brief research monograph uses modern mathematical methods to investigate partial differential equations with nonlinear convolution terms, enabling readers to understand the concept of a solution and its asymptotic behavior. 

In their full generality, these inequalities display a non-local structure. Classical methods, such as maximum principle or sub- and super-solution methods, do not apply to this context. This work discusses partial differential inequalities (instead of differential equations) for which there is no variational setting.

This current work brings forward other methods that prove to be useful in understanding the concept of a solution and its asymptotic behavior related to partial differential inequalities with nonlinear convolution terms. It promotes and illustrates the use of a priori estimates, Harnack inequalities, and integral representation of solutions.

One of the first monographs on this rapidly expanding field, the presentwork appeals to graduate and postgraduate students as well as to researchers in the field of partial differential equations and nonlinear analysis.

“This is a very interesting, brief companion book for anyone interested in nonlocal convolutional operators, including weakly nonlocal diffusion operators, and partial differential inequalities. ... the exposition is very well organized with numerous references supplied, many pertaining to further interesting and engaging problems.” (Joseph L. Shomberg, Mathematical Reviews, April, 2025) 

“This book is a self-contained study of partial differential inequalities with nonlinear convolution terms approached by modern mathematical methods. The motivation of this work is to illustrate how techniques including a priori estimates, Harnack inequalities and integral representation of solutions are used to this type of inequalities … . This book includes in a unitary presentation some author’s original research published in the last few years on the topic … .” (Gabriela Marinoschi, zbMATH 1542.35001, 2024)

Tartalomjegyzék:

Chapter 1. Preliminary Facts.- Chapter 2. Quasilinear Elliptic Inequalities with Convolution Terms.- Chapter 3. Singular and Bounded Solutions for Quasilinear Inequalities.- Chapter 4. Polyharmonic Inequalities with Convolution Terms.- Chapter 5. Quasilinear Parabolic Inequalities with Convolution Terms.- Chapter 6. Higher Order Evolution Inequalities with Convolution Terms.- Appendix A. Some Properties of Superharmonic Functions.- Appendix B. Harnack Inequalities for Quasilinear Elliptic Operators.- Bibliography.- Index.